Can I fire an add_action hook inside of a function that recieves data via wp_ajax?

I guess you have a small misunderstanding here (a slight inconsistency only) add_action is to register a callback for a hook-name, you actually want to fire that hook (not register it).

You can fire a hook with the do_action function.

The Codex-page of the add_action function outlines both:

And naturally the do_action Codex-page has all the glory details:

Take care you don’t create an endless loop, that will crash PHP and therefore WordPress but it also shows that there are no limitations built in.

You can stack as many actions as you want to, even recursive action calls are possible.

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