They don't.
And that clearly not what many of these challenges are "asking" for.
They do, however, have the "power" to make a determination as to what the Constitutional requirement for POTUS, "Natural Born Citizen", means (i.e. what the framers intent was)...and therefore who may be considered a NBC for POTUS eligibility.
If they determine that someone born with foreign allegiance owed is not eligible, then it would be up to someone else to "remove" him (presumably law enforcement since impeachment doesn't apply to usurpers) if he didn't remove himself voluntarily.
Law enforcement
would be shredded by minigun fire should they attempt such.
Only Congress can remove a president. It says so in the Constitution. What part of sole power
do you not understand?