[ChromeVox] Convert ISearchHandler to TypeScript

AX-Relnotes: n/a.
Bug: b/267329383
Change-Id: I5ab7bdb3f75978d263596420f46e84d35e0927e2
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5600080


Commit-Queue: Anastasia Helfinstein <anastasi@google.com>
Reviewed-by: Kyungjun Lee <kyungjunlee@google.com>
Cr-Commit-Position: refs/heads/main@{#1312932}