With Causes, constituents can easily contact, demand action from, and hold their lawmakers accountable. Causes allows you to put petitions onto the site, get people to sign your petition, create and share videos, and take action. 

The original petition site, Care2 offers free and sponsored petition options to help you influence decision-makers and grow your list. 

Action Network is a fantastic petition tool for your site with a growing suite of other campaign tools (think mass emails and surveys) built for progressive organizations and campaigns. It is great for collaborative, coalition-based efforts. Action Network is also embeddable, so you can keep your branding consistent.