Question

As a coequal branch, I can't imagine why the House can't get the President's tax returns as long as they have a legislative or investigative purpose (which they clearly do). That's not to say they could grab anyone's tax returns of any private citizen for no reason.

In the end, I think the Supreme Court would decide, assuming the President resisted. IMHO, the Supreme Court would rule 9 - 0 that the House can see them.

if the house voted to obtain tax returns due to an ongoing investigation that they could go directly to the IRS. Maybe just wishful thinking.

(among others) can demand them from the Secretary of the Treasury pursuant to subsection (f) of the following statutory provision:
https://www.law.cornell.edu/uscode/text/26/6103

Of course, while the statute says the Secretary of the Treasury "shall" provide the returns, you can bet your bottom dollar that Steve Mnuchin will ignore or refuse to comply with the request, setting up a court battle.

with a legitimate reason

What will likely happen is a legal challenge over the House's right to request them.