Looking for a cheap VHS we bought in the late 80's or early 90's. Had rabbit character I believe named Tiggy Furrtail. I'm sure of the name, not the spelling? She wouldn't go to sleep without it. She is 26 now and would like to find it it or info.

This is the only thing that came up when I searched.
https://www.google.com/search?q=audio+talking+books.net&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a&channel=fflb

Hmmm any chance it was an episode of Kissyfur? A show from the 80's... the main character was actually a bear (Kissyfur), but there was a white girl bunny, with red bows on her ears named Beehunnie... here's the intro and ending:
Kissyfur - Opening & Ending - YouTube (http://www.youtube.com/watch?v=tZY6tO97qj0)

Try KidsTube (http://www.kidstube.com) Sedo Domain Parking - Sedo GmbH (http://www.dhgates.com) and Welcome to Tiggy's Rib-Ticklers! The Best in Jokes And Humor (http://www.tiggysribticklers.com)

Nohelp - I tried all your links, one you have to log in or sign up to use, one is something about domain names, and the last one has this disclaimer: Tiggy's Rib-Ticklers are the spiciest humor in town. If you are under 18, or easily offended, this isn't the site for you. For cleaner humor, try Dobhran's Humorama! or Humor Haus!
How is any of that related at all to the op's question?

In the search engine they said they were tiggy furtail cartoon archives

Nohelp - it says Tiggy's Cartoon Archieves. I always double check my links after I post them just to make sure they go where I hope they do. Anyhow, I double checked on the Kids Tube info you posted, I just entered 'kidstube Tiggy Furtail' - there is a video available to watch without having to sign up or log in, my apologies. Here it is, hopefully the op comes back to verify:
KidsTube - The New Teacher (http://kidstube.com/video/69672/The-New-Teacher)

You are right I should have double checked. I got off the computer before I had a chance to. I will remember to double check from now on before something else distracts me.