sp_blitzFirst Does Not Output to DB When Run as SQL Job
Moving the answer from the comments:

When this happens, start by double-checking the contents of your Agent job, and make sure it exactly matches the command you're running manually.

In this case, the text of the Agent job was calling for a different database name - one that didn't exist - so it was skipping the logging portion. Right now, if you call it with a database name that doesn't exist, it doesn't fail hard. It probably should - if you want to file a Github issue for that at http://FirstResponderKit.org, I can add that.