It's called "committee access" and has been around since the 20's (but not used very much)
Chair of the House Ways and Means Committee (or Senate Finance Committee, Joint Committee on Taxation) can request the tax returns. No actions happens in Congress as that request goes straight to the IRS (specifically, Mnuchin, the Treasury Sec.).
From there (here's where the "not used very much" part comes in), it's sort of unknown as committee access has never actually been in front of a judge so it's likely Trump uses the full Justice Dept. to fight the request from the House.
If the House wins, his tax returns would almost certainly see the light of day. There are privacy protections, but whomever ends up with his returns could simply make up a legislative purpose (i.e. possible conflicts of interest) and make them available to the full House (or Senate). From there you can almost guarantee they end up online.
https://taxprof.typepad.com/files/154tn1013-yin.pdf