It's estimated that about 84 million workers are affected by the requirement.
Some labor unions have also filed legal challenges in support of the rule, and some contend that the president's mandate doesn't go far enough to protect workers from the virus.
Workers protest a coronavirus vaccination requirement at aviation company Boeing, on a street in St. Louis, Mo, on October 20. Photo by Bill Greenblatt/UPI
Under federal law, each U.S. Circuit Court that's received a case will get one entry in the lottery. The entries will be placed in a drum, and a clerk at the Washington, D.C., Judicial Panel on Multidistrict Litigation will randomly draw one.
Supreme Court Chief Justice John Roberts appointed the members of the panel, which is comprised of several federal judges from different circuits.
Proponents on both sides of the issue are hoping that a court friendly to their ideologies gets to decide the matter.
"While the Republican-led states have filed in courts with a majority of judges appointed by a Republican president, the unions are targeting courts with a majority -- or near majority -- of judges appointed by a Democratic President," attorney Sean Marotta told CNN.
Republicans are hoping the cases fall in a conservative-leaning court like the 5th Circuit Court of Appeals, which extended an injunction against enforcing the vaccine mandate while the matter makes its way through the courts.