[react-sync] Open pull requests as the bot that authors the commits (#96682)
Noticed when auto-merge didn't work. `vercel-release-bot` isn't in the
exemption list (for good reason).
Since `vercel-release-bot` still uses a static token, it's time to use
the app fully.
The rest of the usage I'll do in a follow-up.
## test plan
It'll fail for now when enabling auto-merge. Everything else works
though. I'll fix the perms for auto-merge in a follow-up.
- [x] [sync from this
branch](https://github.com/vercel/next.js/actions/runs/30954178134/job/92143210455)
-> https://github.com/vercel/next.js/pull/96688