Where do hiri and ottappa (shame, and fear of wrongdoing) fit in?

The purpose of the shame and fear of wrongdoing is to promote virtue (sila) and good conduct.

From the Noble Eightfold Path, Right Action, Right Speech and Right Livelihood comprise the portion on virtue (sila). The Five Precepts are also for virtue.

The Noble Eightfold Path comes from the fourth noble truth of The Four Noble Truths i.e. the path to ending suffering.

From AN 5.57:

"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.’

As you have quoted correctly, AN 7.65 says that hiri and ottappa are important for the practice of virtue (or ethical conduct):

"Mendicants, when there is no conscience and prudence, one who lacks conscience and prudence has destroyed a vital condition for sense restraint.
“Hirottappe, bhikkhave, asati hirottappavipannassa hatūpaniso hoti indriyasaṃvaro;

When there is no sense restraint, one who lacks sense restraint has destroyed a vital condition for ethical conduct.
indriyasaṃvare asati indriyasaṃvaravipannassa hatūpanisaṃ hoti sīlaṃ;

Upasaga Frank K, intetested,

- Namo tassa bhagavato arahato sammā-sambuddhassa -

☝ Shame and fear of wrong-doing requires much care how you look (in the eyes of wise) 💎

As for The Noble Eightfold Path (ariya-magga)

Hiri & ottappa (shame & fear of wrongdoing) is an aspect of right resolve, going after the desire of not harming, causing no conflicts and restrain, renouncing (sense desires, desire after becoming). Hiri & ottappa are working here toward development of the virtue section and having the purification of virtue as result, resulting then in further refining through right effort (again a pattern of hiri & ottapa), driving one into the heart of the conncentration section. As Nyom Rubben mentioned, they work for the perfection of Sila as prerequisite of right samadhi and panna.

In The Four Frames of Reference (satipaṭṭhāna) they work at all for in the area of ardent, alert, & mindful — putting aside greed & distress with reference to the world.

The Four Right Exertions (sammappadhāna), they cover the hole drive for right effort.

Likewise in The Four Bases of Power (iddhipāda) on a refined level.

In The Five Faculties (indrīya), Sila and it's cause is found in The faculty of conviction (saddhindrīya) and faculty of persistence (viriyindrīya).

Likewise in The Five Strengths (bala) as The strength of conviction (saddhā-bala) and strength of persistence (viriya-bala)..

In The Seven Factors for Awakening (bojjhaṅga) they are included in Mindfulness as a factor for Awakening (sati-sambojjhaṅga), Analysis of qualities as a factor for Awakening (dhamma-vicaya-sambojjhaṅga), Persistence as a factor for Awakening (viriya-sambojjhaṅga).

So hiri & opatta are factors supporting Conviction: Right Speech (Eightfold Path), Right Action (Eightfold Path), Right Livelihood (Eightfold Path), Desire (Bases of Power) and Persistence: Right Effort (Eightfold Path), Four Right Exertions, Persistence (Bases of Power) Persistence (Factors for Awakening).

As coming from Right resolve, effort, bases on right view, hiri & opatta has right view as for-runner:

There is the case where a certain person is not covetous. He does not covet the belongings of others, thinking, 'O, that what belongs to others would be mine!' He bears no ill will and is not corrupt in the resolves of his heart. [He thinks,] 'May these beings be free from animosity, free from oppression, free from trouble, and may they look after themselves with ease!' He has right view and is not warped in the way he sees things: 'There is what is given, what is offered, what is sacrificed. There are fruits & results of good & bad actions. There is this world & the next world. There is mother & father. There are spontaneously reborn beings; there are brahmans & contemplatives who, faring rightly & practicing rightly, proclaim this world & the next after having directly known & realized it for themselves.' This is how one is made pure in three ways by mental action."

One lacking in right view may think before or/and after deeds: to deny responsibility, to deny that harm was actually done, to deny the worth of the victim, to attack the accuser, and to claim that they were acting in the service of a higher cause.

And the Buddhas cut offs, maintained by right view are: (1) We are always responsible for our conscious choices. (2) We should always put ourselves in the other person's place. (3) All beings are worthy of respect. (4) We should regard those who point out our faults as if they were pointing out treasure. (5) There are no — repeat, no — higher purposes that excuse breaking the basic precepts of ethical behavior. (see Reconciliation, Right & Wrong, in regard of changing ways)

Usual sense-desire and desire for becoming (joy, happiness) are the "higher pruposes" why people lose shame and fear of wrongdoing.

(right) Hiri & Opatta are governed be the desire for becoming and to a state even beyind becoming, to leve behind low stands, states and enter that of the Noble Ones, governed by one or all three governing principles (right view):

"There are these three governing principles. Which three? The self as a governing principle, the cosmos as a governing principle, and the Dhamma as a governing principle. ...Adhipateyya Sutta: Governing Principles

As there is shame and fear of wrong doing actually present in all beings and actions, but for the most shame in doing good and fear to lose face under fools, this talk may be usefull supply to leave associations with fools toward wise:

In the Eyes of the Wise: The Buddha’s Teachings on Honor & Shame

May no one be ever ashamed in doing merits and may no one fear to do wrong what fools regard as their refuge, to find real happiness for oneself with ease.

A possible extended answer, with further cross-links and given space for discussion and further investigation can be found here:

[Q&A] Where does hiri & ottappa (shame & fear of wrongdoing) fit in the Path?

(Note that this is not given for trade, exchange, stacks or entertainment but for development of restrain from unskillful toward liberation and make meritorious deeds with and on it)