The most popular homeowner gunstock finish is probably Tru-Oil. You can get it at almost every store that sells guns.
Tru-Oil is polymerized linseed oil. It is linseed oil that has been cooked at high temperatures in an oxygen-free environment (inert gases) until it just begins to gel, then rapidly cooled. The cooking causes the carbon atoms to begin crosslinking. So when the finish is later exposed to oxygen in the air during the finishing process, the finish cures very rapidly.
After the cooking the oil is very thick so the manufacturer adds mineral spirits to make the finish easier to apply. You can add more mineral spirits if you like.
Tru-Oil is oil, but it acts more like varnish. It can even be built up like varnish, but only with very thin coats or it will crack. Gun owners like to “rub in” the finish anyway, even though no more finish enters the wood after the first coat has dried, and this removes most of the excess. So thick finishes aren’t generally a problem.
Let each coat dry hard (a few hours in a warm room), sand off any dust nibs if there are any, then apply more coats until you get an appearance you like. If the finish is glossier than you want, dull it by rubbing with #0000 steel wool.