Formatter Action doesn't post comments when made from a fork

When a PR is made from a fork the Formatter failed by doesn't post comments with clickable suggestions:
see: https://github.com/JuliaDiff/ChainRulesCore.jl/pull/488/checks?check_run_id=3854350212#step:5:187

reviewdog: This GitHub token doesn't have write permission of Review API [1],
so reviewdog will report results via logging command [2] and create annotations similar to
github-pr-check reporter as a fallback.
[1]: https://docs.github.com/en/actions/reference/events-that-trigger-workflows#pull_request_target,
[2]: https://help.github.com/en/actions/automating-your-workflow-with-github-actions/development-tools-for-github-actions#logging-commands

I am not sure if we can raise the permission on the GITHUB_TOKEN on a fork,
or if we can issue another token that has that permission?

But checks that fail without being easy to act upon suck.
This isn't super hard to do locally, I think it is just

using Pkg: @pkg_str
pkg"activate --temp"
pkg"add JuliaFormatter"
using JuliaFormatter; format("."; verbose=true)

But that is still work.

And especially for PR from newcomers (not @mcabbott 😂 ) who will always be making a PR from a fork, we want things to be smooth.

Start with the Formatter Action configuration and the linked check's reviewdog output, then reproduce formatting locally with the JuliaFormatter command shown in the issue. Trace how fork pull requests receive tokens and report results; done means determining a safe way to post clickable suggestions, or documenting the supported fallback if that is not possible.