Sometimes. but even when they do, it's hard to make the case that they worked. Perhaps it was something that was going to happen anyway.
And as Veganite points out, "...but what's better, saying nothing at all? "
One sort of general rule is that petitions work better with businesses than with government. When a government is elected they already feel they are acting according to the will of the people. (that might be incorrect - but that's the theory)
History is strewn with examples of failed petitions. Maybe the most famous example occurred in the 1800s in the UK where people were petitioning for a number of reforms including universal suffrage. The petition had to be rolled onto the Parliament floor and contained 3 million signatures. It was rejected.
Here in the USA there have been many petitions for increased gun control. Not much has happened. but President Obama did ask Congres to pass laws after Sandy Hook, that probably was in response to the millions of signatures on petitions.
One of my favorite petition stories is about Charlie the dog. He was in a leash free park when he ran after a police horse. the horse got spooked and reared and the policeman fell off. Then the horse ran away and got bruised up some. The pound was notified and the dog was captured and was scheduled to be destroyed. San Franciscans mobilized and 10,000 signed a petition. Supposedly the fastest 10,000 signatures ever obtained in SF. The dog was taken off death row and sent to dog obedience school and finally fostered out. but probably more due to the $20,000 collected in the legal defense fund and a good lawyer.
Its sort of thought that companies respond to petitions better than government. but it's not always so. I signed petitions, with millions of others to get FOX to renew Firefly and to get CW to renew Veronica Mars. but neither petition got the network to change its mind.
If you join a group like the Sierra Club, PETA, WWF, they regularly send out newsletters with pleas for signatures. It only takes a minute. I also subscribe to Care 2 Healthy Living which sends out a newsletter that has links to online petitions.
However, taking a lesson from Charlie the dog. The 10,000 signatures might have helped. but it was the $20,000 legal defense fund that paid for a good lawyer that saved his life.
FYI
https://www.bustle.com/p/petitions-that-actually-changed-the-world-34309
https://www.huffingtonpost.com/2014/02/25/white-house-petitions-works_n_4848866.html
https://sanfrancisco.cbslocal.com/2...ath-sentence-after-crissy-field-horse-attack/