TL;DR version

There is a conditioned and compounded, dependently originated self-doer, which is a discernible endeavouring and initiating being, that is generating the volitions to perform action. But there is no standalone independent eternal entity that is acting as the driver or charioteer.

An example is the autopilot of a self-flying airplane. It's a self-flying airplane, not a Self flying airplane. Several computers together with software, sensors and actuators in the plane work together to calculate, decide, and instruct how the plane flies. There is no human pilot (who is independent of the airplane) who is flying the self-flying plane. The autopilot is very much an integral and inseparable part of the airplane, just like its landing gear, engines and wings.

Similarly, there is no independent Self or Atman working like a "human pilot" performing action. Rather, there is a dependently originated self-doer working like the autopilot which is calculating, deciding and performing the actions. The self-doer is very much an integral and inseparable part of the five aggregates.

If one thinks that nobody is doing anything, this will result in an unwholesome attitude of not doing anything or not taking responsibility, which in turn results in unskillful actions, which in turn results in deepening suffering. For more info on this, please refer to the commentaries by Nizamis and Piya Tan below on fatalism.

Furthermore, to think there is no self at all, is an annihilationist view, which is a false view, just like the eternalist view.

Long version

The Attakari Sutta (AN 6.38) answers this:

Then a certain brahman approached the Blessed One; having approached the Blessed One, he exchanged friendly greetings. After pleasant conversation had passed between them, he sat to one side. Having sat to one side, the brahman spoke to the Blessed One thus:

“Venerable Gotama, I am one of such a doctrine, of such a view: ‘There is no self-doer, there is no other-doer.’” (Note1)

“I have not, brahman, seen or heard such a doctrine, such a view. How, indeed, could one — moving forward by himself, moving back by himself (Note2) — say: ‘There is no self-doer, there is no other-doer’? What do you think, brahmin, is there an element or principle of initiating or beginning an action?”

“Just so, Venerable Sir.”

“When there is an element of initiating, are initiating beings clearly discerned?”

“Just so, Venerable Sir.”

“So, brahmin, when there is the element of initiating, initiating beings are clearly discerned; of such beings, this is the self-doer, this, the other-doer.

“What do you think, brahmin, is there an element of exertion ... is there an element of effort ... is there an element of steadfastness ... is there an element of persistence ... is there an element of endeavoring?”

“Just so, Venerable Sir.”

“When there is an element of endeavoring, are endeavoring beings clearly discerned?”

“Just so, Venerable Sir.”

“So, brahmin, when there is the element of endeavoring, endeavoring beings are clearly discerned; of such beings, this is the self-doer, this, the other-doer. I have not, brahmin, seen or heard such a doctrine, such a view as yours. How, indeed, could one — moving forward by himself, moving back by himself — say ‘There is no self-doer, there is no other-doer’?”

“Superb, Venerable Gotama! Superb, Venerable Gotama! Venerable Gotama has made the Dhamma clear in many ways, as though he were turning upright what had been turned upside down, revealing what had been concealed, showing the way to one who was lost, or holding up a lamp in the dark: ‘Those who have eyes see forms!’ Just so, the Venerable Gotama has illuminated the Dhamma in various ways. I go to Venerable Gotama as refuge, and to the Dhamma, and to the assembly of monks. From this day, for as long as I am endowed with breath, let Venerable Gotama remember me as a lay follower who has gone to him for refuge.”

Commentary by Nizamis:

Note1: “Natthi attakāro, natthi parakāro.” Some people might have expected the Buddha to have approved highly of this naïve negative doctrine. The fact that he very succinctly and effectively refutes it is extremely instructive and of great significance for gaining a better understanding of the depth, subtlety, and holism of the Buddha’s actual teaching. Although the Buddha taught that there is no permanent, eternal, immutable, independently-existing core “self” (attā), he also taught that there is “action” or “doing”, and that it is therefore meaningful to speak of one who intends, initiates, sustains and completes actions and deeds, and who is therefore an ethically responsible and culpable being. It should be quite clear from its usage in this sutta, and from the argument of this sutta, that kāra in atta-kāra must be an agent noun, “doer, maker”: this is strongly entailed, for example, by the Buddha’s statement: “ārabbhavanto sattā paññāyanti, ayaṃ sattānaṃ attakāro ayaṃ parakāro”, “initiating beings are clearly discerned: of (such) beings, this is the self-doer, this, the other-doer” (AN iii.338). (This is perhaps even clearer than the term hāra in bhāra-hāra meaning “bearer” (“burden-bearer”) in SN 22.22 (Bhāra Sutta: The Burden; PTS SN iii.25). SN 22.22, which describes the “bearer” of the “burden” of the “five clung-to aggregates” (pañc-upādāna-kkhandhā) as the “person” (puggala), is arguably very closely related to AN 6.38 in meaning and implications. See SN 22.22 and also SN 12.61, note 1.) Atta-kāra could mean that one motivates oneself, or that one acts upon oneself; para-kāra could refer to the atta-kāra as seen from a third-person perspective, or to one who acts upon another being or thing. In each one of these cases, there is necessarily an all-important moment of initiation of action. As for the form of the term atta-kārī, which occurs in the title of this sutta, compare the expression: “yathā-vādī tathā-kārī”, “one who speaks thus, one who does thus”; or, in other words, “he does as he says”, “he practises what he preaches” (compare, for example, PTS DN iii.135, AN ii.24, Sn 359).

Note2: Sayaṃ abhikkamanto: “moving forward by oneself”; sayaṃ paṭikkamanto: “moving backward by oneself”. Sayaṃ means “self; by oneself”. The example seems to suggest the action of someone who intentionally takes a step forward, and then intentionally takes a step back again. This example leads directly to the next statement, and thus emphasises the idea of initiating an action: when someone takes a step forward or backward, the origin and impetus of this action must, so to speak, come from “somewhere” or “something”. In other words, it really is an intentionally initiated action (kiriya, kriyā), and not merely the arbitrary mechanical “effect” of some prior mechanical “cause” in a deterministic chain of mechanical push-and-pull. The sense, here, can be better understood if one also consults AN 2.33 (Aññataro Brāhmaṇo: A Certain Brahmin; PTS AN i.62), where the Buddha describes himself thus: “I am one who asserts that which ought to be done... and one who asserts that which ought not to be done.” (“Kiriyavādī cāhaṃ... akiriyavādī cā'ti.”) There, it is made very clear that doing and non-doing are morally significant and morally effective. Similarly, in several other suttas, such as in MN 95 (Caṅkī Sutta; PTS MN ii.164), the Buddha is described thus: “The venerable recluse Gotama is truly one who asserts the doctrine of kamma, one who asserts the doctrine of what ought to be done. . .” (“Samaṇo khalu bho gotamo kammavādī kiriyavādī. . .” (MN ii.167).) Again, in MN 71 (Tevijjavacchagotta Sutta; PTS MN i.481), the Buddha humorously recounts that in the last ninety-one aeons, no ājivaka, or “fatalist” who denies the power of volitional acts, has ever gone to heaven, except one, who happened to follow the doctrine of kamma and of morally effective deeds (“sopāsi kammavādī kiriyavādī”, MN i.483).

Part of Piya Tan's commentary on the Attakari Sutta:

2 Causality and agency

2.1 The Attakari Sutta is a short discourse on causality and agency, and is a clear statement on the efficacy of personal effort. Here, causality refers to a karmic action we (or anyone) have initiated, and such an action has karmic consequences. This is only one “link” in a complex karmic network of multiple causes and conditions. For, all actions work in concert with other actions, producing a network of results. This is called conditionality.

2.2 Agency, on the other hand, refers to how an action or set of actions we do affects others. For example, we hear a Dharma talk (action), and inspired by it, we go on to meditate (result). The Dharma speaker is an “agent,” so to speak, to our being motivated to meditate. A better Buddhist explanation is that the agency here is the teaching (not the person) of the Dharma teacher who motivated us. In short, it is possible to move others to goodness. It is for this reason that the Buddha teaches us the Dharma: we can be moved to act in a way that would transform us into awakened beings.5

2.3 The efficacy of action view (kamma,vāda) is that our present condition is partly the result of our deeds in previous lives. Those who reject this view, that is, the non-efficacy of action (akiriyā,vāda), believe that all deeds, past, present or future, have no effect on the condition of beings. “Thus, in its extreme forms the [akiriyā,vādi] would say that there is actually no causal connection between what a living being does and what he is or becomes, in this or in another life” (Gomez 1975:81 f).

2.4 The teaching on the efficacy of action is mentioned in greater detail in the Sonadanda Sutta (D 4.6), where the Buddha declare himself to be “one who teaches karma, who teaches (the efficacy of) action” (kamma,vādi kiriyā,vādi) (D 4.6/1:115). Similarly, in the Anna,titthiya Sutta (S 12.24), Sāriputta declares to the wanderers that the Buddha teaches the efficacy of action (S 12.14/2:33 ff). A threefold classification of non-action (akiriya) in the Titth’āyatana Sutta (A 3.61): our present condition is all due to our past actions; that it is the result of a god’s creation; or, that it is by sheer chance (A 3.61.1-4).6

2.5 Strictly speaking, the Atta,kārī Sutta is not about free will, but rather about fatalism. Philosophically, free will can only operate in a deterministic situation. If we do A, then it always follow that B will occur. If the situation is indeterministic, we will have no way of knowing how our actions will result. Fatalism, on the other hand, is an ethical view that choice is meaningless.

2.6 Makkhali Gosāla’s niyati,vāda is fatalism, which the Buddha rejects because Gosāla claims that purification happens without cause or condition within the individual (D 2).7 Furthermore, like the brahmin in the Atta,kārī Sutta, he claims that there is neither human agency (atta,kāra), nor effort (viriya), nor human exertion (purisa,parakkama) (id).8 Here, we are dealing with the ethics of what we can or cannot do.

2.7 As scholars of philosophy, like Asaf Federman, have noted, “The Buddhist rejection of this view is not a rejection of deterministic theory of causality but a rejection of fatalism.” (2010:13).9 It should be noted that the Atta,kārī Sutta is not discussing philosophy, but rejecting the view (especially that of Gosāla and of the brahmin in this Sutta) purporting or suggesting that we are all fated, so that no effort possible for spiritual purification and liberation. The Buddha teaches the contrary: we can help ourselves. Indeed, if we do not, no one can help us (Dh 160, 380).

The permanent, separate self does not exist. If there is no one there, who does the doing?

We can not control our five senses and thought. I do not choose the colors of the trees, the sounds of a guitar, the taste of carrots, or the preferences I have. These are all uninfluencable, yet ever changing. If all the faculties of the mind are beyond our control, how could we have the power to do anything?

Our 5 senses and thought are all manifestations of awareness. Without awareness of our sight, there would be no sight, etc. Therefore, our five senses are made of nothing but awareness. They aren't even separate from awareness, they are nothing but awareness. Things are not happening. Awareness is being.

Reality is not what you infer from your senses. The universe is not outside of your mind. Reality is the awareness projecting your mind. The universe is the whole of which your mind is derived. There is no doer, only one being that knows nothing of doing.

Here is a quote from a little story I wrote several years ago about Gautama's path to Buddhahood. In my story, this was the Buddha's answer to Kaundinya (Koṇḍañña) asking for the main point of Buddha's teaching:

"It is possible to think that we are merely will-less bodies, aggregates of physical elements, not having the free will, ruled by the laws of nature. It is possible to assume that we are the accumulations of memes, informational viruses, while our bodies are only their physical media. It is possible to consider that we are simply humans - suffering, dying, performing skillful or unskillful acts. It is possible to identify oneself with the eternal and infinite Totality of All. It is possible to believe in life after death, or in complete annihilation, or in perpetual transformation. It is possible to see people's minds controlled by the gods and demons, or by the rumors possessing the crowds. There is no single formula providing the answers to all the questions.

Do not rush to conclusion: "I get it now, the Teaching of Buddha is agnosticism". All I said above is not about that, but here is what it is about:

There exist an infinite variety of perfectly valid, mutually exclusive, but internally consistent, perspectives onto the world. Each of them leads to different decisions, different actions, and different results. You should decide for yourself - what to believe in and how to see the world. Do not let anyone force their perspective on you, however logical and experience-based it would seem. Do not take anything for granted without critical consideration. Because every time you're letting in a thought, you are choosing your path, and not only yours.

You can choose to think such thoughts from which you will suffer yourself and will torture others. Or else, you can choose to look at things such that neither you will suffer, nor the others. But there is no single point of view that would fit all the situations in life. A point of view is like a tool - use the right tool for each job.

Remember: by accepting some point of view you are changing not just your personal future, but also the future of the entire world. It depends on you what it will be. Choose your reality well."