Is there a specific time that you can start using TMs again if you've hit your daily limit, or is it a rolling 24 hr period?

Rolling. As actions expire 24 hours after happening, they drop off of the count. Once enough actions drop off and your action total goes below 50, you will be able to use TMs again. Please keep in mind that this count is NOT the action count that shows on your ship. This count includes disables, hacks, raids, invasions, and artifacts used against a player or planet.

Thank you for a thorough explanation. Exactly what I needed to know.