I can't find anything about it on the page devoted to accessing Audible content on a Kindle Fire:
https://audible-uk.custhelp.com/app/answers/detail/a_id/8807/related/1
However these is a glimmer of hope on the FAQs page:
https://audible-uk.custhelp.com/?pf_rd_p=2449fbd7-6dcb-4ed3-9eff-3aca883774fc&pf_rd_r=BVZMTCR5J3CRS6EQXAHW
(If that link doesn't work, go here
https://www.audible.co.uk/ , scroll to the foot of the page and click on 'Help').
Click on 'How do I buy audiobooks on an iOS device?'. (Yes, I know that a Kindle Fire isn't an iOS device but stick with it anyway).
That allows you to access a new link, labelled 'Purchasing from the mobile website'. There you can see how to download books via a web browser, rather than through the Audible app. The instructions conclude with this:
"Scroll down the page of your audiobook and tap either Buy With 1 Credit or Buy for £X.XX"
So that, contrary to everything else on the Audible site, suggests that it
is possible to purchase books without using Audible credits (for which you need an account). However you'll probably still need the app to actually play the recordings.