When Congress votes on something, the vote cast by every representative is made public. Is this required by law? Specifically, is either chamber of Congress allowed to hold a binding vote without making the votes of each member known to anyone beyond the leadership?
It's not possible for a citizen to cast an informed vote, if he/she doesn't know the voting record of the candidate. So, I would say, there's no way members of Congress can conceal any of their votes.
In terms of the supporting documentation you're requesting, here are three sources:
Record of Proceedings. Each House shall keep a Journal of its Proceedings, and from time to time publish the same, excepting such Parts as may in their Judgment require Secrecy; and the Yeas and Nays of the Members of either House on any question shall, at the desire of one fifth of those present, be entered on the Journal.
So, at the most basic level, the Constitution requires both chambers to publish a record of members' votes ("Yeas and Nays").
Rules of the House of Representatives (pdf, 47 pages)
Five methods for members to cast their votes are listed in Section XX. Voting and Quorum Calls (page 33). None of the methods allow for members' votes to be concealed.
All voting in Congress is a matter of public record.
Congress itself doesn’t even have to know which members voted which way. Both houses allow for voice votes, which is where the presiding officer says “all in favor say yea... all opposed nay...” and decides which side won based on who seemed to have more people. The Constitution only requires that members cast recorded votes if a fifth of those present want it done.