Git Reports is a free service that lets you set up a stable URL for anonymous users to submit bugs and other Issues to your GitHub repositories, without exposing your code, requiring users to log in with GitHub, or using a messy two-repository solution. It works with public and private repositories, and it works with personal and organization repositories. It's super easy to set up: log in, allow access, and activate your repository from the list! Check out the tutorial for more details.
Free and ad-free! This service is the side project of a broke college student, so if you like it, throw me a few dollars.
The GitHub API doesn't have a whole lot of scope options; unfortunately, in order to get write access to a repo's issues, I also have to request write access to the full repo. The site's open-source, though, so you can check it out for yourself or even self-host your own copy if that's your thing.