New mutants start popping up all over and everyone wants in on the action. Who shows up at the homes and places of business trying to recruit (or capture) them?

Obviously the X-Men are going to be there. Who've they got opposing them? Is there still a Brotherhood of Evil Mutants? Some incarnation of the Hellfire Club, perhaps... Mister Sinister, of course. And A.I.M. might be interested in the science of weaponizing new mutants' powers, and Hydra would likely be interested in gaining some new recruits... I suppose if the individual had a spectacular public display of power for good or bad, the Avengers might show up... and/or S.H.I.E.L.D. If the mutants' manifesting aren't overly noticeable, someone might know someone who tries to hook up a group of thugs looking for an edge in their next heist.

Am I missing anyone?

There's likely still sections of the Weapon Plus project in operation, with hidden labs and other facilities. They're big on wanting to turn mutants into weapons.

The High Evolutionary, if he's off on one of his tangents involving mutants.

Apocalypse
Mojo
Project Pegasus
Possibly a current incarnation of the Sentinel Program

Mutant Liberation Front

Really underutilized group/concept/story-line.