Please note that compatibility for Kindle Fire is sometimes reached by a workaround, as not all library apps have been allowed into the Amazon App Store. 

Don't know what generation of Kindle Fire you have?  The 1st Generation have an orange "unlock" tab, or see this chart from Love My Fire.

Services available:	What is it?	Versions compatible?
RBdigital	Audiobooks - books you listen to.  Mostly fiction & some popular nonfiction. App is actually the same version for regular Android tablets, so instructions are combined.	3rd Gen (2013) & newer - app in your Amazon App Store. App will not appear in the store or install from incompatible Kindles.
cloudLibrary	eBooks - mostly fiction, some popular nonfiction. Bigger publishers & bestsellers.  Browse, checkout, and reading are all done inside the app.	2nd Gen & newer, with workaround.
Enki Library	eBooks - wide variety of fiction & nonfiction which also includes independent publishers & self-published books. Set up a reader app, then use the website to checkout & download a book.	All, with workaround to load Aldiko Reader.
Freading	eBooks - wide variety of fiction & nonfiction, very large selection.  No holds. Set up the Freading reader app, then use the website to checkout/download the books into the reader.	All, with workaround to load Freading app.
Total Boox	eBooks - wide variety of fiction & nonfiction, large selection. No holds, checkouts do not expire. Browse, checkout, and reading are all done inside the app.	Any newer than the 2012 HD 8.9 version, app in your Amazon App Store.
Britannica E-STAX	Kids' nonfiction eBooks - no specific checkout, use as many books as you would like. Read in your browser, please note an offline use app is not available for Kindle Fire.	All
BookFlix	Digital storybooks - No checkout, use as many books as you like. Books are viewed in browser so no apps or downloads of files.	All

Kindle Fire 1st Generation - now too old for some apps.
You'll note from above it can still use:

• Enki eBooks - the Aldiko Reader app works well even on older devices.
• The children's materials: BookFlix & Britannica E-STAX.
• RBdigital audiobooks - your device is too old for the app, but try the Listen Online feature.