[ChromeVox] Convert PermissionChecker to TypeScript

AX-Relnotes: n/a.
Bug: b/267329383
Change-Id: I209535f3d6538d54ae5362b7750cc9b6d29faf70
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5545729


Reviewed-by: Jacob Francis <francisjp@google.com>
Commit-Queue: Anastasia Helfinstein <anastasi@google.com>
Cr-Commit-Position: refs/heads/main@{#1313075}