Can you play an action if an opponent conquers an open battlefield?

Yes, you can play an action if an opponent conquers an open battlefield.

According to RiftJudge FAQ #5832 and RiftJudge FAQ #2682, moving a unit to an open battlefield creates an open showdown. During an open showdown, all players may play both [Action] and [Reaction] cards.

Timing Sequence

When an opponent moves a unit to an open battlefield, the following occurs:

1. Showdown Opens: A non-combat showdown begins (Rule 344.2).
2. Focus: The opponent (who applied the Contested status) gains focus (Rule 345).
3. Playing Cards: In an open state, the player with focus may start a chain by playing an [Action] or [Reaction] card.
4. Responding: Once a chain is started, the state becomes "Closed." Players may add new items to the chain using [Reaction] cards (Rule 358.3).
5. Resolution: Per Rule 340.1, the newest (top) item on the chain resolves first (Last-In, First-Out).
6. Priority Windows: According to RiftJudge FAQ #10001 and FAQ #4385, after each individual item on the chain resolves, there is a window of priority where players can add new [Reaction] cards to the top of the remaining chain.

In summary, because the showdown is open, you have the opportunity to respond to the move with [Action] spells to potentially prevent the conquer or alter the board state.

Note: I could not fully verify this ruling against the retrieved rules.