If I play an Ambush unit to Mystic Vortex, does it count as a reaction card and I have to pay more cost for that?

Yes. If you play an Ambush unit to a battlefield where you control Units during a showdown at Mystic Vortex, the card has Reaction as a characteristic (granted conditionally by Ambush), and the cost increase of [A] applies.

Here's why: Ambush (Rule 822.1.b) grants the unit "I have [Reaction] as long as I'm being played to a battlefield where you control Units." Mystic Vortex checks whether cards have the Reaction characteristic (Rule 813.5), not whether the word is printed on the card. Since the Ambush condition is met, the unit has Reaction at the time of play, and Mystic Vortex's cost increase is applied during Step 3 of determining total cost (Rule 356.3).

This is consistent with how Mystic Vortex treats Quick-Draw and Hidden cards — it doesn't matter where the Reaction comes from; if the card has it, the tax applies. (FAQ #11686)