Dresden Files plays with this a little. Rock salt shotguns, silver bullets, and garlic/holy water paintball guns all make appearances.

I like the idea, and it is possible. There's a couple of options for firing a wooden projectile. Guns exist that use .22LR blanks to fire an arrow from a traditional rifle. Shotgun shells seem like the next best option, however I don't know how well they would do on penetration. Probably similar results to less-than-lethal rounds like rubber pellets or pepper balls. You could fashion a pointed slug, but in that case, you could do that for most any bullet.

I think theres a better argument to be made for silver bullets. They're already metal, not sure if silver is harder or softer than lead though. You could also do a salt shot through a shotgun. Salt is supposedly effective against undead, and there's that fun tidbit about vampires being hella OCD and they'd have to count all of the salt before they can do anything else.

I think the best would be a shotgun that shoots 00 buckshot made of silver, but specifically something like the smith&Wesson governor, or the Taurus judge. They're handguns, specifically revolvers, that are capable of shooting both 45 long colt, and 410 shotgun shells without any modification needed for the gun. Fuck it, add some garlic salt to your shells as well. Garlic burns them, and it would potentially smell good too.

This has been a very fun little thought exercise. If you come up with any other sci-fi/paranormal gun related questions, I'm more than happy to participate

Fiocchi makes shotgun loads with rubber projectiles. You could do the same with wooden projectiles.

I'm not an airsoft person, but I assume wood pellets would work in those guns, too.

If you want longer range then we can start playing with the rules of your hypothetical situation. Does the wood have to physically contact the vampire? With some effort, you could make a bullet with a wood core and a lead or brass jacket.

12 ga. Saboted wooden slugs? That'd probably be my approach though I suspect you'd still struggle with getting enough penetration.

Video on wooden slugs: https://youtu.be/hVrN3f1Ce5w

You could also shoot stakes via pneumatics.

I think the critical question is how much wood is necessary? If a small amount of wood will work then you could take an existing bullet with a ballistic tip and just replace the tip with a piece of hardwood.