[Extensions] Use promises in Search API test.

References: crrev.com/c/2368147 & testing_api.md#use-promises.

Bug: 1119846

Change-Id: Id7e603c415f8d2c6fee00a615b97078e486fe6f7
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2382888


Commit-Queue: Solomon Kinard <solomonkinard@chromium.org>
Reviewed-by: Devlin <rdevlin.cronin@chromium.org>
Cr-Commit-Position: refs/heads/master@{#805595}