This is a REALLY sill question, that my husband asked me and I don't know the answer.
With mink bears, the ones that I have seen are made from reclaimed mink. Can you also get mink for bears etc that is made in a similar way to the alpaca and mohair, where no animal is harmed in the making?
You would never be able to shear a mink. Mink have a nasty temper and sharp teeth. They would shred you to pieces if you tried. Even if you wore heavy gloves and a mask, you would never be able to hold one down long enough to shear it.
Like Joanne said, it would be too hard to make usable yarn for weaving out of mink hair. The hairs are too short and stiff to card and spin into yarn.
You're better off using the whole pelt. Recycling old fur coats is a fine way to make a Bear. Just be sure that the pelts are not too old and stiff. Mink coats do wear out. Use good quality pelts if you want a good quality Bear.
You might be able to get some scraps from a furrier. Just about every furrier has a mink Teddy Bear or two in his shop. You might even be able to strike a deal. The furrier gives you some fur to work with, either at a discount or for free, and you promise to make a Bear that he can sell in his shop.
Secondarily, real fur pelts are somewhat trickier to work with than plush fabric. If you are able to cultivate a relationship with a furrier both of you could benefit. He gets Bears to sell in his shop and you get to learn how to sew real fur.