Can I use an action card after a unit has been gusted from the battlefield and there are no more units there?

Yes — you can still play an Action card. The Showdown continues normally until both players pass Focus on an empty chain, even if your unit was Gusted away and no units remain at the battlefield (FAQ #1570; Rule 348).

During the Showdown Open state, the player with Focus may play Action or Reaction cards regardless of unit presence — e.g., you could play Ride the Wind to return your unit, or any other legal Action spell. You also don't lose control of the battlefield while a Showdown/Combat is ongoing there (Rule 323.6), so playing a unit back to that battlefield remains legal.

The window only closes once both players pass Focus: the Showdown closes, and a Combat Showdown then proceeds to resolution — where, with no units, you'd lose the combat.