I've sewed-up a ton of red fox furs over the years for a local buyer and my own when I was shooting them with the 22-250....
A SIMPLE HOLE is fairly easy to sew up, IF the edges are pretty smooth and can easily be layed together in a natural manner.
BULLET HOLES are another story. Usually the bullet has struck a bone, creating a ragged, jagged exit hole. Those edges should be removed, rounded-off and just made to go together smoothly before sewing is ever attempted. You will have a lot stronger and better looking "fix" if you spend the time taken to smooth off the hole.
Barring the loss of an extra large patch of fur, it is usually possible to "put together" a fur that has been pretty shot up and sell it for little or no loss in monies.
Any hair or fur you have sticking through to the inside of the hide should be carefully pushed back through to the fur side. If you don't do it and have a considerable amount of fur on the inside, it can/will be noticed easily they the trained-eye of a good fur buyer!
The better, smoother job you do of sewing, the less attention the fur will get at the buyers shop.
That lead me to use any smaller-sized, regular sewing needle and a light (hide colored) regular sewing thread from my wife's sewing box.
A single strand of thread, carefully pulled up so the hide-edges were very close together and then tied-off worked for me.
Afer a loop through the hide and a small over-hand knot to secure the bitter-end of the thread, there are a few stitches that work especially well for sewing up furs so they won't pull out.
One of the best is what I call a "Cable Lacers Stitch". The needle is ran through the torn hide on one side, about 1/8th to 1/4-inch from the edge of the tear, then over to the other side of the tear, through that side and then back to the inside of the hide. Instead of starting another stitch at this point, run the thread underneath the thread as it closes the space between the two pieces of hide. That simple step will "lock" each stitch in place.
Don't pull your stitches up tight!!!! Just snug them up so the hide touches edges and leave it alone. Smooth up the completed sewing job with your finger pressure on the hide to even out any irregularities.
The main thing about sewing hides it to take your time. A nasty hole can usually be "fixed-up" pretty good if you do.
For the heavier furs such as coon and coyotes, you CAN use heavier or doubled thread. But for the lighter hides, stay with a single, smaller thread....
When you turn the fur, carefully brush-out the area where the hole existed. It should be barely noticeable, IF at all....
It works for me!!!!!