Go to your management and let them know. Your co-worker is using company time and resources to promote himself to a political office which is almost definitely against their policy, or at least is improper workplace behavior if not outright harassment. You may phrase it diplomatically:
My coworker has repeatedly asked me to sign a petition to promote his political campaign. I am uncomfortable with people pressing their politics on me in the office. I don't wish to turn this into a major incident, but perhaps my coworker's management could let him know that his actions are not universally appreciated and might be against company policy.
Probably nothing will be done right away, but your complaint will at least be noted, and if your coworker retaliates against you for refusing to sign his petition, you will have put in place a basis for him to be fired. "Remember when I came to you last month? Just as I feared, he became enraged when I did not sign his petition and has been saying nasty things about me in the lunch room."
It's unfortunate that some people are too self-centered and narcissistic to realize the impact of their actions on others. It's doubly unfortunate that you have to work with this person. Your first loyalty is to yourself and the security of your job and livelihood, so take appropriate preventative measures now and not become a victim of his bullying later on.
I have seen this kind of behavior many times in my career and rarely do people respond well to moral persuasion in the workplace. Usually, it is only the threat to their direct self-interest, e.g. their job, that convinces them to back off. However, usually employers don't have the patience to be forgiving, unless he's some kind of super-star employee whom they can't afford to fire, and those are very very rare!