In general do you need to register or anything like that? I think it'd be a fun thing to try, but also don't want any of the bad legal repercussions that can come with it
All bug bounty programs have rules that outline what parts of their site/product you can test and what kinds of bugs they are looking for (here's Facebook's https://www.facebook.com/whitehat/). As long as you follow the rules you won't have any legal problems.