Each area animal shelter is different in what they want or can take, with that said all you have to do is call and ask them.
The shelter I give too, will take anything that can be washed. Old towels, blankets, quilts just anything that the dogs and cats can lay-down on. I have taken two old hand towels (or 1 bath towel cut in half), sewn both sides and bottom shut. Then stuffed with cut up t-shirts, towels and scraps. Then sew the end shut. I stuff as full as I can, because cats and small dogs like the nest thing. For the larger animals (mostly dogs, but sometimes small pigs and goats too), I have used an old towel that was stained but still good as the middle of my pet quilt. But most of the time I use POLYESTER (you know from the 60's and 70's - that stuff that NEVER dies and the color never fades), I always have yards around (because someone has cleaned out mom or grandmas house and they used to sew). I take this and use it in place of batting, it doesn't breath so it helps get animals warm. They can dig or chew and it will not hurt it. I have even used it as the backing on blankets not stuffed.
My animal shelter is so HAPPY when they are given any kind of bedding for the animals. Last year when I was cleaning out the house after my mom passed, I was unable to think (it was a hard time). So as I cleaned I packed stuff for my self and other family members, got stuff ready for garage sale and donations. I started with donations (this was the first, the items I didn't want to sell or give to family or friends), they went to the local animal shelter thrift store. The first trip was a truck full (all cleaned and ready to be priced and sold), I went back twice more that first day. The thrift store is used to fund foster homes for animals, low cost animal care of all kinds and it helps keep the kill numbers down in the area. Before it was all said and done, I made 21 trips to the thrift store (small truck not full sized). Between each trip (as I started to feel better, I would sew dog blankets and beds), I was also able to drop them off at the thrift store for the shelter (shelter is out of town). I felt better because of being able to help the animals (and my folks would have liked that), so a hard time was made easier by using my sewing and quilting skills to help the animals. This spring I opened the mail and found a very sweet note from the manager of the thrift store, telling me thank you again for the donations to the store and shelter. I was stunned, but it was nice to know that these items had helped. On adaption days, the shelter send the dogs home with the blanket they had with them. Shelters are always good places to give to.