When Mellie is filibustering the spending bill, the Vice-President comes in, takes over the Chair, and proceeds to ask a question (to give Mellie a break). However, the President of the Senate (or the officer presiding) is not allowed to participate in debates, a rule that goes back to the first Congress. The Vice-President would not be allowed to ask a question during the filibuster.
The writers made an odd choice having Mellie filibuster an abortion-related bill (making funding for Planned Parenthood discretionary), as this is something the Senate Democrats would certainly also be agitating about, whereas the show depicts them as not caring.