For Linux-related service requests, we first ask that folks try an Ubuntu 22.04 or Fedora 36 Live USB (the distros we have done the most internal testing with and created setup guides for) to be able to determine whether there could be a hardware issue. Once we have verified there isn't a hardware issue, we ask that folks post in the community thread for their distro for help: https://community.frame.work/c/framework-laptop/linux/91
In practice, this works well because we have an extremely helpful and engaged community (including in many cases maintainers for that distro). Additionally, because that debugging happens in the open, any answers from it are publicly visible for future users to see.
All of that said, we'd love to find better ways to provide deeper support ourselves and are open to input. A more official path would likely still start with the most popular distros.