Andrew Liptock started the petition, citing the fact that the Rams are 0-8 in games officiated by Vinovich.
Liptock also pointed out that the opposing team has never been assessed more penalty yardage than the Rams in games overseen by Vinovich's crew.
Liptock's petition has already garnered more than 1,000 signatures toward a goal of 1,500.
According to J.B. Long of ESPN Los Angeles, the Rams' 0-8 record in games refereed by Vinovich has been accrued since 2006.
This season, the Rams lost games to both the Saints and Philadelphia Eagles with Vinovich's crew in charge of the action.
The 45-35 loss to the Saints in Week 9 featured a questionable call that saw Rams punter Johnny Hekker get called short of the first-down marker on a fake punt:
That call was a potential momentum-shifter, as the score was tied 14-14 when it happened.
In the Week 9 meeting between the Rams and Saints, the Rams were penalized four times for 32 yards, and the Saints were penalized just twice for 20 yards.
As part of his argument to remove Vinovich, Liptock also noted that he is the only official to ever eject current Rams defensive tackle Aaron Donald and former Rams defensive end Chris Long.
Assuming Vinovich and his crew remain in place, they will be on hand at the Mercedes-Benz Superdome in New Orleans on Sunday to witness who will represent the NFC in Super Bowl LIII.