What are the possibilities? I’m sure they will do what they can do.
1. Go to court. That's where disputes between the Legislative and the Executive are usually resolved. The 1st Circuit, however, is a mixed bag and it would be quite possible to end up with a judge who won't recognize an open and shut case of Administration lawlessness.
2. Assign the House Sergeant-at-Arms to arrest whoever refuses to respond to the Committee's subpoena and seize the evidence requested. It is unclear, however, whether the House actually retains this weaponry -- it hasn't been deployed for eighty years (not since 1934).
3. Take House action to defund something critical to the Administration, pending a satisfactory response. Such action, however, would require the Speaker to stand for the Committee and its investigatory objective. Not a certainty, in my mind.
Perhaps, there are more. And I would suggest that Chairman Gowdy is investigating them all.