No, the rule given in the 12th amendment is that a majority of the whole number of Senators is needed, so that's 51 Senators since we have 100 Senators. Even if a Senator isn't there for the vote so that there is a 50-49 majority in the vote that's not sufficient.
The Vice President has no role in selecting his successor beyond voting in November.
(There's also the awkward possibility of 34 Republican or Democratic Senators refusing to show up so that there is no Quorum, but I don't think we've reached that level of incivility just yet.)