It depends on what you mean by "effective". The vote to elect House Speaker is done in a closed door election, so we the people won't get to see who votes what, so there is no direct connection between a petition and this vote.

However, politicians are reluctant to do unpopular things, so stuff like a big petition or wide scale protests or even lots of people being interviewed saying the same thing can set the stage for politicians to feel comfortable going against the party apparatchiks. Hakeem Jeffries is a deeply political creature who has benefitted in the past from being popular in his area, but with all the hullabaloo being made about his feckless nature, his career is in deep trouble. He doesn't have much more social capital to spend and if he isn't careful, the gravy train he's been enjoying for so long will come to an end.

This is the exact situation he was trying to avoid in the midterms. Because of this blue wave, the Dems have a lot of power they can exert in Congress. However, Jeffries doesn't actually want to do his job, so he spent a lot of time and effort trying to depress Dem voter turnout during the midterms. He is still trying to dampen enthusiasm to excuse not doing his job in the future.

If Jeffries is smart, he'll quietly step aside to salvage what remains of his career. If he can read the room, he should see that the petitions and protests and interviews are signs that public sentiment has turned against him. He would have a reduced role, and probably a reduced amount of bribes, but he'd still have a career.

If he's stupid, he'll try to continue leading the party to sit on their hands. A delusional man would fight against the wave and, unfortunately, probably do a lot of damage while trying to have his cake and eat it too. There is no way he would be able to thread the needle of, "do just enough to look competent" and "do nothing enough to keep getting bribes" and eventually lose his job.

If he's a fuckin asshole, he'll try to sabotage the Dems as much as he can to go out in a final blaze of glory. Shamelessly wreck things while taking money from anyone interested in hurting the Dems and then run away with his ill-gotten gains instead of running for re-election. Probably to Isreal.