Yes, file the copies with your stock purchase/vesting agreement. Also if you have paid par value or whatever amount for the shares keep your canceled check with

Yes, file the copies with your stock purchase/vesting agreement. Also if you have paid par value or whatever amount for the shares keep your canceled check with the copy of the shares you bought. The company keeps a list of who owns what shares as well as the physical shares. If something should happen to the physical shares they can be reissued.