Your theme is probably forcing the use of wpautop() on the content area of your posts in some way. wpautop() is the function WordPress uses to convert the line breaks in your posts to
My suggestion is to try switching your site to another theme for a quick test. You can switch to one of the default themes that WordPress includes, like "Twenty Twelve". Once you've switched themes, reload your petition page and see if the
<p> tags problem in the petition has gone away.
If it does go away, then the theme you were using is at fault. You can decide if its worth changing to another theme permanently or if you would prefer to just turn off the "Allow custom messages" option for your petitions. Petitions without that option turned on shouldn't have the visible
<p> tags problem even if your theme is doing something strange with wpautop().