You can vote on anything you may choose to vote on, I suppose. The question is whether that's the appropriate route to take when the information you're seeking pertains to a particular bill before the House. One might suggest you vote against the legislation.
On the other hand, you may vote against the legislation and still think your privileges are being breached, and you still may want to pursue the privilege point because there's a principle here. What you want to do is move this matter, I suppose, to the point where the House affirms its rights. Whether the House takes any further steps beyond that, that's another matter for the House to consider.