Well, I can tell you how Not to do it, from experience many years ago.
Don't think you'll get away with just filling the holes with wood putty, as it makes a perfect little black dot the second you stain it. Also, don't even consider using wood toothpicks as you'll get the same black dot, it just takes a little longer. Nor use any wood, where the end grain of the wood is at the surface level,,, same ol' black dot.
One way to do it is to first fill most of the hole with a filler (toothpick) leaving it about 1/16" below the surface of the stock. Then find a piece of wood to match that of the stock, get the grain running the same direction as that of the rifles, then cut tiny pieces to plug into the tack holes, being careful to maintain the correct grain direction. Those you can make about as invisible as possible!
PS, If the tack head has left a mark you can try to steam that out or sand it out if it's a newish gun. But you'll be stuck with a lot of refinishing then.
John