You can't.

You can use your credits from your subscription, but you cannot pay for a new audiobook in the app.

They changed this recently, and do allow direct in-app purchases now. In some random podcast interview with an exec at Amazon, I remember them saying that each team gets to make their own independent decision on this, and Audible as a unit decided to allow in-app purchases and pay Apple their cut.

Yes you can. I just did. And it was cheaper than the listed price on web.

It’s a relatively recent feature (past year), but it’s possible.

Because the Audible unit at Amazon set their own policy on this, and decided it was better for them and their users to pay Apple's 30% cut.

No idea why the price through the app was better than the price on the website. Maybe some kind of promotion? Weird.

Someone in Amazon probably decided that Audible's unit economics could afford to take a 30% cut, while Kindle's could not.

I'm not surprised, since you can often buy a Kindle + Audible book for cheaper than buying the Audible book alone.

If you look at the Audible book (such as Wool https://www.audible.com/pd/Wool-Audiobook/B0BKR3Y6SP) the price is often higher than if you look at the Kindle book price, and adding the Audible narration (https://www.amazon.com/Wool-Hugh-Howey-ebook/dp/B088SY4GSD).

In the Wool example, the Audible book is $31.64, but the Kindle book with the audible add-on is $9.48 ($1.99 + $7.49). Even if the Kindle book was selling for "list price", it would still be cheaper at $27.48.

So, I'm guessing there's just a lot less margin headroom to eat a 30% cut on Kindle prices, but plenty to eat that on Audible's prices.