The Volokh Conspiracy
Mostly law professors | Sometimes contrarian | Often libertarian | Always independent
As Akhil Amar has pointed out, either House can propose constitutional amendments. Since 1808, amendments can address any topic under the sun. So the House might take an interest in constraining the President, or expanding his powers, or granting itself plenary power, or reallocating Senators (among consenting states), or establishing a church, or reopening final bankruptcy judgments, or regulating the private lives of celebrities, or whatever.
Right now, Congress's Article I powers are limited. The failure to respect those limits doomed the SG's argument in Lopez. But Congress has the 'horizontal' necessary and proper power, to make laws for carrying into execution the President's powers—and it has powers under Article V too. So it's hard to find a topic categorically beyond any joint resolution the House might adopt, or beyond its potential need for relevant information.
Of course, there might be limits other than topical limits. Maybe needing the information for something isn't enough. Maybe exploratory interest doesn't count, without specific plans or proposals under review. Maybe there are limits of good faith, or of executive privilege, or on judicial enforcement, or on the House's power to investigate in general, or on its capacity to appear in an Article III court, or …. But legislative topic seems the wrong place to look, for the parties and the Justices both.