Constitutional crises don’t come any bigger than one branch of government denying another branch its basic ability to function. So if the Veep were to assert a right to be able to keep the Senate from convening, I doubt there would be much discussion over the minutia of the Senate rules of order. I think it’s much more likely that the Capitol would be ringed by tanks and legislators would be gasping for breath in clouds of tear gas.
But, I will point out a couple of things. When the Senate moves to adjourn (which they must do any time they’re not in session, even just until the next morning), that motion will specify the date and time when the Senate will reconvene. At that time, the presiding officer will gavel the Senate to order. Until that gavel bangs, no member of the Senate can assert any rights or really do anything official – until they are convened, they’re just a bunch of guys standing around on the Senate floor. So the Veep could refuse to call the Senate to order, but there’s really nothing he could do to prevent the President Pro Tem or his designee from doing so.
Once the Senate is convened, a Senator MIGHT be able to raise a point of order that the President Pro Tem did not have the standing to call the Senate to order. The Veep cannot do this (he cannot offer motions or even vote on anything unless it’s a tie) so he would need an ally in the Senate. The Chair (which would be the Veep – once the Senate is convened, he’s free to tell the President Pro Tem to get out of his chair) would rule on the point of order, presumably upholding it. However, the Chair’s ruling is subject to an appeal by any Senator, with a simple majority needed to overturn the Chair’s ruling.
But again, this is getting into territory that the rules never anticipated, and my basis for the above procedural outline is on a bit of shaky ground. My best advice to legislators in this scenario remains to bring their gas masks.