How does one go about removing a stripped compression plug? I'm assuming it needs to be tapped? Job best left to a bike shop?

Depends if it's a star nut or expansion plug.

Starnuts can sometimes be broken, or turned sideways and eased back up. generally it's easier to push them deeper and out of the way and use a new starnut.

Expansion plugs should be disassembled and the top part lifted out, then the expanding sleeve or shoes can often be eased out by sticking a finger in and working them up. Many of the types that only have a single split are difficult to lift out this way, so I use an old screwdriver that I bent into an "L" or open "J" shape years ago to get behind them and pull them up. Lastly turn the fork over and shake the expanding cone out.

The one thing you usually can't do with expansion plugs is pull or push them out assembled, since any pressure applied to the expanding cone expands the sleeve and locks them in.

It's an expansion plug. It has 4 splits in it. Not sure what you are saying is a viable option in my case.

Expansion plugs are all the same in that pulling them up while assembled pulls the expansion cone on the bottom up into the wedges jamming them.

Sometimes you can lift them really gently but usually it's a self locking action and will defeat you. remove the top unit, and let the cone fall to the bottom, then lift the expanding section out by finger or hooked tool. If your plug has the expanding section and top unit all in one piece it's still the same, loosen the expander cone until it drops then pull out the expanding section.

Well thats the problem. The "bolt" in the center is rounded out. I cant get a tool in there to loosen or tighten it.

I assume you're talking of the bolt that controls the expansion, and not the top cap screw. If that's right, drill down through it and the bottom cone will be free, then you can pull the expansion plug out in sections.

Yes thats correct.