[OOBE] - Assign callback before sending any requests to the server

Assign callback before starting Almanac connection so won't get into
this code path twice if data is requested concurrently by two clients.

Bug: b/345502399
Change-Id: I9dd700c34f8e3e17e2ed88b96644dedba13c2e19
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5614791
Reviewed-by: Danila Kuzmin <dkuzmin@google.com>
Commit-Queue: Bohdan Tyshchenko <bohdanty@google.com>
Cr-Commit-Position: refs/heads/main@{#1312843}