A petition is a supplication, request or prayer and has this meaning in a spiritual sense. Nowadays it has the additional meaning of a request from like-minded people.
Psychological / emotional perspective: It is worthwhile deciding which is more important in the dream, the petition itself, the cause or the people with whom we are working. We can then consciously decide where our loyalties lie in what is represented.
Material aspects: To be signing a petition indicates committing to a deeply held belief.
To refuse to sign means we question our commitment.
To have a petition refused signifies disappointment in a working environment.
Whether you filled it out, or were getting sign-ups, to dream about a petition means that you are making progress toward your goals.