Responsibility in Buddhism

If nothing can be considered 'myself' or 'mine', if nothing is in my complete control (take volition for example), how can people be held responsible for their thoughts, words and deeds, if they are by definition not in charge?

This question hints at two things to my mind: the notion of free will or agency and the notion of karma as a judge of right and wrong. Further, I think there might be confusion in the question with regards to the doctrine of anatta and the implication that persons lack agency.

Buddhism does not teach that persons lack volition... exactly the opposite! Buddhism teaches that actions are only accomplished with intent and persons manifestly do have agency. If persons did not have agency and we were all zombies forced to follow a completely deterministic path, then obviously there would be nothing to do. What could we do? Nothing. What choice would we have? None.

This is manifestly not what the Buddha taught.

Karma is also not some sort of judge of our right and wrong actions punishing us for our evil deeds and rewarding us for our righteous ones. The law of karma is simply an observation that certain actions bare certain fruits and we classify accordingly.

The doctrine of anatta and sunyata is deep and difficult to comprehend. It is famously said the Buddha despaired and hesitated before teaching it fearing that no one would be able to understand it. I think this saying is a precaution to guide us away from thinking the doctrine of anatta or emptiness somehow implies that persons lack agency or that our actions do not matter. Nothing could be further from the truth.

When the Buddha taught about "not-self", he did so as a "realisation". This means, in the realised mind/person, selfishness ends; thus unwholesome & irresponsible actions end.

The Buddha did not teach "not-self" as expressed in the question. In other words, to laypeople living ordinary lives, the Buddha taught about "self" & "kamma" (i.e., "moral responsibility"). The suttas about "kamma & re-birth" do not teach about "not-self" ("anatta").

MN 117 describes the two sorts of Right View taught by the Buddha: (i) worldly right view for worldlings; and (ii) Noble Right view for aspirants. Kammic inheritance is worldly right view and anatta is Noble Right View. MN 117 is a crucially important sutta.

Your question is a variation of the question, "if there is no self, then who is responsible for actions?"

The Buddha rubbished the notion that there is no self doing anything, in the Attakari Sutta:

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

“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 — 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’?”

There is a self (the self-doer), but this self is impermanent, conditioned and compounded. It arises from the inter-working of the five aggregates, and ceases when the five aggregates cease. By breaking down a violin to its constituent components, you cannot find something called music. Similarly, by breaking down the five aggregates, you cannot find anything called the self. The self, like music, depends on the inter-working of other components, that are themselves dependent on other things. The self, like music, is not non-existent. Please see this answer for more details.

There is such a thing as the concept of wholesome thinking in Buddhism. Being virtuous, one reaps material benefits (see Ittha Sutta) and is also lead towards enlightenment (see Kimattha Sutta).

So, how can a lay person cultivate a wholesome way of thinking that would lead him towards becoming virtuous? AN5.57 holds the answer:

“And for the sake of what benefit should a woman or a man, a householder or one gone forth, often reflect thus: ‘I am the owner of my kamma, the heir of my kamma; I have kamma as my origin, kamma as my relative, kamma as my resort; I will be the heir of whatever kamma, good or bad, that I do’? People engage in misconduct by body, speech, and mind. But when one often reflects upon this theme, such misconduct is either completely abandoned or diminished. It is for the sake of this benefit that a woman or a man, a householder or one gone forth, should often reflect thus: ‘I am the owner of my kamma, the heir of my kamma; I have kamma as my origin, kamma as my relative, kamma as my resort; I will be the heir of whatever kamma, good or bad, that I do.’

And what about a noble disciple?

“This noble disciple reflects thus: ‘I am not the only one who is the owner of one’s kamma, the heir of one’s kamma; who has kamma as one’s origin, kamma as one’s relative, kamma as one’s resort; who will be the heir of whatever kamma, good or bad, that one does. All beings that come and go, that pass away and undergo rebirth, are owners of their kamma, heirs of their kamma; all have kamma as their origin, kamma as their relative, kamma as their resort; all will be heirs of whatever kamma, good or bad, that they do.’ As he often reflects on this theme, the path is generated. He pursues this path, develops it, and cultivates it. As he does so, the fetters are entirely abandoned and the underlying tendencies are uprooted.

By reflecting on the notion that all beings are owners of their kamma (and inherit the results), the noble disciple cultivates a wholesome thinking that results in renunciation.