A recognized definition of pull is to do something, as in "pull an all-nighter." It is fairly common slang to use the phrase "pull a [person's name]" to mean replicating some accomplishment of that person. Usually, this accomplishment is at the very least surprising or even some underhanded deception. There's a related idiom, to pull a fast one, which means to deceive or defraud someone.
In this case, "pull a Jefferson" is being used to mean "replicate Thomas Jefferson's actions in 1800 counting the electoral votes." In the Presidential election of 1800, Thomas Jefferson was the sitting Vice President, so he presided over the counting of the votes. He was also a candidate for President and the eventual winner. During the counting of the initial electoral votes, there was an error in the certificate of election from Georgia, although it was clear that the electors had chosen Thomas Jefferson and Aaron Burr. Jefferson counted those votes despite the error.
The writer of the article you are quoting believes that this sets a precedent that the Vice President (Mike Pence in this case), while presiding over the counting of the votes, has the ability to choose accept or reject erroneous slates of electors from a particular state. So for the VP "to pull a Jefferson" in this case would mean choosing not to count the votes of electors from states with supposed fraudulent results.
