What is the best way to get my 93-year-old, sight-impaired, technologically impaired mother set up for listening to audiobooks on her own?

Try Recordings for the Blind through the Library of Congress. (She doesn't have to be blind.) They provide everything for free, and they send a very basic cassette recorder that anyone can easily use. There's virtually nothing to do to play the books except to hit the play button.

https://www.loc.gov/nls/

Are you in the US? If so, call her local library and ask them for the contact information for the state library of whatever state she lives in. Tell the person(s) at the library that you are interested in the Talking Books program and would like to sign her up, There is no charge to her. It is a gracious gift from the taxpayers to help people with visual or any other challenges for whom reading a physical book is difficult or impossible. The library will "permanently" loan her a very robust, not-very-complicated machine which she can use until she no longer needs it, and then you ship it back. The machine is really very easy. the sound quality is extremely clear, and you can adjust the speed of the reading and volume of the reading. The cartridge is big enough to hang onto easily, and there is only one way it will go in, much easier than a CD player. When you hover over each button, it tells you what it does. It remembers where you were in your "reading" and starts at the spot the next time you turn it on. You can tell the state librarian what types of books your mother likes to read, and they send a cartridge with a new book now and then. Returning the cartridge is easy. Simply flip the mailing card on the outside of the specially designed it came in, and send it out in the mail (no postage needed). In addition to this, if you know your Mom's tastes pretty well and you have a computer and are willing to spend a little time, the BARD website (connected to the Library of Congress) has a website from which you can download onto a blank cartridge (really just an easy-hold USB drive that slips into the machine the right way) from thousands of titles. These cartidges hold a lot-- lots more than just one book. These run the gamut as to length and subject--something for everybody. Files and can be added or subtracted from this little drive at will. I have done this off and on for several years for my parents. The file format for these works exclusively for these government-issued reading machines. The blank USB drives can even be bought from places like Amazon, if you ever lost the one you had, or you wanted a spare so you could always have her have something to listen to, and you could play with loading the other one and sending it to her. The machine has a battery with an impressive life, which is recharged once in awhile by plugging it in for a few hours. Or you can just run it with power from the wall and not worry about the battery. If there is ever a problem with the machine, send it back and they send another one. Kindly volunteers do most of the maintenance on these machines, I believe. It is actually almost unbelievable. My parents have had many happy hours with this machine, and the librarian at our state library has always been very gracious and helpful. Best wishes!

If she already has wireless internet, I’d suggest trying one of the Alexa devices. My aunt liked hers until her cognitive issues got too bad to work it on her own anymore. We started her with an echo dot we got really cheap on a promotional offer the moved on to an Echo show 10 so she could accept video calls. (Her vision wasn’t good enough to get much out of the screen herself, but she could hold things up to the camera so we could look at them for her.)

Amazon is best at the functions that make them money. So Audible books are much easier to use than Hoopla library books.

It will work best if you do all the setup ahead of time so she isn’t intimidated: unbox the device, create her Amazon and Audible accounts, gift her a book to try so she doesn’t start with a subscription that you have to cancel, add in a few contacts, maybe change the wake word from Alexa to Computer if she likes the original Star Trek…

Then take it to her place, introduce it to her Wi-Fi, and say:
Computer. Read [book title].
Computer. Sleep in one hour.

There are lots of other things to try with varying success. You can set up the Echo Show to try to identify objects.
Computer. What am I holding?

We did the audiobook ordering for my aunt. She could also call us on her phone and we’d coach her on getting a book stared at bedtime if had trouble.

My Mom preferred books on CD and used one of these. https://smile.amazon.com/gp/aw/d/B0714DX1Z3
Apply CD and pull cord to start. Pull cord again to stop. No sleep function so we used a plug that had a timer.

for my mom’s alexa dot and Show, I have her app on my phone and I set up a routine to play a certain book at a particular time. She gets an alert and then it plays the book (from audible). I do the same for her wake up alarm. I have it set to play her favorite radio station (BBN.org) or BBN radio. It is commercial free Christian music, Bible readings, audio books and general uplifting programming. I can adjust the volume remotely as well as play.

My mom was issued a free NLS digital talking-book player from the National Library Service for the Blind and Print Disabled. Her eye doctor told her about it. I remember filling out a form for her, where she got to choose the type of books to read. They would mail digital books to the house, and she could return them easily . The device was designed to be easy for the blind. The number I found for them is 1-888-657-7323. Good luck!

You're quite sure she wants this, are you? Has she already tried and enjoyed audio books if someone has set them up for her? It's just that, if not, you could be on a hiding to nothing.

Simple recorder for tapes or CDs with easy to use buttons - better if buttons have raised designs to indicate forward, backward, and stop.

Sometimes it is hard for people to learn new skills at a late age. She may need assistance getting the audio started. I got my mother a retro radio with knobs, and with her advanced dementia, she couldn't turn it on herself, but she enjoyed listening to music and we kept it set to a station she liked and her aide turned it on and off.

asfastas1can: Perhaps you can garner assistance from the Commissioner of the Blind for your state of Illinois.

InItForGood - has the perfect answer below.

We used the services for my husband who lost his vision. It is simple and easy to use and there is no charge.

Each state has their own library for the blind. I believe it is sponsored by the federal and state governments.