I am not 100% sure this thread is within the purview of the 2nd Amendment Forum, but since Garland's previous rulings with regard to the 2nd Amendment could be seen as less than favorable to the cause, I supposed it's in the neighborhood.
This is theoretically how it would work, quoted from the petition website. Joe Biden is not spearheading the effort, but rather is the target of the petition.
Sign the Petition to Joe Biden and Senate Democrats - Confirm Merrick Garland to Supreme Court
"At 12:00 noon on January 3, 2017 (according to the Constitution), the terms of 34 U.S. Senators will expire. At that point, the Senate will briefly consist of 66 sitting senators—until Vice President Joe Biden, in his capacity as Senate president, begins swearing in the senators-elect.
Before Biden begins the proceedings, he has a chance to preside over a Senate that consists of 34 Democrats, 2 independents who caucus with Democrats and 30 Republicans—as the remaining Senators are in limbo of being newly sworn in. At this point, Democrats could ask to finish Senate business as it pertains to President Obama’s nomination of Judge Merrick Garland."
The Petition fails to recognize the long standing and highly structured rules of the Senate for the first day of a new Congress, as set forth by the Senate itself. Any change would have required agreement by the previous Congress, which did not happen.
The general procedure, a small part of which is quoted below, is found here:
The First Day of a New Congress: A Guide to Proceedings on the Senate Floor
This is the last schedule, from 2015, but the principles remain the same.
"The Constitution mandates that Congress convene at noon on January 3, unless the preceding Congress by law designated a different day. P.L. 113-201 set January 6, 2015, as the convening date of the 114 th Congress.
etc
etc
The important part is that the new Senators are sworn in before a quorum is established.