The last-ditch legal effort, filed Sunday, came from Gohmert, an eight-term congressman from Texas, along with 11 Arizona residents who had been nominated by that state's Republican Party to serve as electors.
It comes over a week before Pence is scheduled to preside over a joint session of Congress where the Electoral College votes for Biden and President Donald Trump will be tallied up.
Electors had already cast their votes two weeks earlier. Biden received 306 electoral votes — 36 more than he needed to win — while Trump received 232.
The suit asks federal Judge Jeremy Kernodle, a Trump appointee in eastern Texas, to declare that Pence has the "exclusive authority and sole discretion" to decide which electoral votes from a given state should be counted.
While pro-Trump electors in some states Biden won have symbolically cast their own ballots, experts say those votes carry no legal weight.
The Republican complaint claims that part of the 1887 Electoral Count Act should be declared unconstitutional as it clashes with the 12th Amendment.
That amendment contains "the exclusive dispute resolution mechanisms," the lawsuit claims, including that "Vice-President Pence determines which slate of electors' votes count, or neither, for that state."
Legal scholars quickly dismissed the Republicans' lawsuit as hopelessly far-fetched.
"No, this won't work," tweeted election law expert Rick Hasen at the University of California, Irvine.
"The suit will go nowhere," wrote Joshua Geltzer, executive director of Georgetown University's Institute for Constitutional Advocacy and Protection.
"This is insane," tweeted Georgia State University law professor Anthony Michael Kreis.
Spokespeople for Pence's office did not immediately respond to CNBC's request for comment.
The suit also asserts that "public reports" have "highlighted wide-spread election fraud" in battleground states, citing a document written by White House advisor Peter Navarro that includes numerous claims that have been rejected in other lawsuits or debunked by fact-checkers.
Trump has refused to concede to Biden. He has falsely asserted he won the race while publicly pressuring Republican lawmakers to "step up and fight for the Presidency." At the same time, Trump is spreading unfounded and debunked conspiracy theories alleging widespread voter and electoral fraud.
Some House Republicans have stated they will challenge the election results when Congress meets to count the electoral votes on Jan. 6. Senate Majority Leader Mitch McConnell, R-Ky., has reportedly urged his caucus not to make similar objections.
Any objections to the electoral votes must be submitted in writing and signed by at least one House and one Senate member. If an objection arises, the two chambers consider the objection separately.
The Trump campaign and several of the president's allies have launched dozens of attempts to challenge the election results in numerous swing states. None of those legal efforts has succeeded in invalidating votes for Biden or flipping the results of any states' presidential elections.
The U.S. Supreme Court earlier in December rejected a bid from Texas Attorney General Ken Paxton to sue four key swing states over changes they made to voting procedures. Trump had called that lawsuit "the big one."