Petitions and the like are outside the scope of this site. But the first step, which is in the scope of this site, is to know what law you want to change. This is likely going to depend on the animal and the state, and possibly local laws as well.
For example, Wisconsin law makes it illegal to possess a "wild animal" under most circumstances without a license. However, there is already an exception for opossums, among other animals.
Under Wisconsin law, "Wild animal" is defined as "any animal of a wild nature that is normally found in the wild and that is not a domestic animal". "Domestic animal", in turn, is defined as "a farm-raised deer, a pet bird, a farm-raised game bird, or an animal that is listed as a domestic animal by rule by the department of agriculture, trade and consumer protection." So, presumably, you'd have two choices if you wanted to legalize an animal not currently exempted: you could get the state legislature to add an exception to the law, or you could get the department of agriculture, trade and consumer protection to modify their rules to get it defined as a domestic animal.