He currently is shooting his moms benelli nova compact 20 guage and it seems to fit him well for now but I know he is also a growing 15 year old. Will this gun continue to work for awhile?
Yes, the shotgun he's currently shooting will work for the time being. My almost-14YO no longer needs a shortened stock; he's as tall as I am, already. My guess is, that your 15YO will need to move up from the "compact" model sooner than later, but otherwise there's nothing wrong with learning the basics with what you have available.
I noticed all the other kids on the team were shooting 12 guage. Will 20 be a handicap for him?
A 12ga is pretty standard for trap. Yes, you give up a little with a 20ga, compared to the 12ga, but at the 16yd line a 20ga can break targets consistently.
I wouldn't worry about getting a dedicated trap gun, at this stage in his shooting career (save your money for ammo). A good field shotgun is just fine, and a semi-auto has less recoil than a pump. A gas-operated 12ga semi-auto would probably recoil about the same (or less) as the current lightweight pump he's shooting now.
The Remington 1100/1187 and Win Super-X1 are solid and fairly affordable choices. I have seen a few in the $400 - $600 price range, recently. I picked up an 11-87 for my son last year.
Most folks are pretty agreeable to letting young shooter try their shotguns, for a round or two. Have him shoot some other shotguns, so he can see what he likes and what fits him better.