In summary, yes, vote-a-rama can continue indefinitely. To take your questions in turn:
- Does every single one of the 100 Senators have to be present and agree to stop the vote-o-rama?
Not actively - that is to say there is no vote on cloture which requires unanimous consent. Vote-a-rama continues as long as there are amendments left to be considered. A sufficiently determined Senator could therefore, in theory, keep vote-a-rama going indefinitely by continuing to offer amendments.
Other Senators are not required to be present, although, for a roll call vote to take place, a 'sufficient second' is required. This is defined as a fifth of the smallest possible quorum - 11 Senators. It's this voting process that actually takes up most of the time - limited to ten minutes. Without a sufficient second, amendments can only be identified and explained, which only takes a couple of minutes, making it a lot more difficult to continue vote-a-rama.
- If not, can Senators take turns to be present in the chamber (while the rest take a break) to offer amendments to let the process continue indefinitely?
Yes. In fact, this was attempted by Sen. Ron Johnson, who told reporters that he was setting up a shift schedule to coordinate efforts to delay the bill:
"Historically what’s happened is ... we offer a couple of hundred
amendments on the Republican side," Johnson said. "And we get a couple
of dozen voted on, and people tire out. I’m coming up with a process
that keeps people from tiring out. I’m getting sign-ups. I’m laying
out a three-shift schedule."
In previous vote-a-ramas, Senators have voted right at the start of one roll-call vote, then left the chamber for a break, returning to vote right at the end of the next roll-call vote, giving themselves about 20-25 minutes break between votes.
- If so, what recourse can the Senate take? In other words, is there any
mechanism that allows the Senate to vote to end the vote-o-rama by a
majority or 3/5 vote?
Not while it's ongoing. Congress' ability to set its own rules itself is guaranteed by the Constitution, but to do so would require the passage of a law such as the Congressional Budget Act of 1974 which created vote-a-rama, or the use of the nuclear option to change Senate rules. Neither of these can take place while vote-a-rama is still taking place.
As Zach Moller, a former Senate Budget Committee staffer put it: "Vote-a-rama is the punishment that the minority party gets to inflict on the majority party for using the streamlined processes set up in the budget process and reconciliation".
