Can you deploy in a showdown?

No — Deploy doesn't let you play gear in a showdown. Deploy is a passive/triggered keyword that only changes where the gear can go (any battlefield) and adds "when an opponent holds here, kill this" (Rule 830.1.b.1); it grants no timing permission, so a Deploy gear defaults to your Main Phase in a Neutral Open state and can't be played once a showdown is underway (Rule 343.1).

To play a Deploy gear during a showdown it needs Action or Reaction — either printed on the card, granted by something like Jarvan IV, Martial Paragon ("Your gear with [Deploy] have [Action]"), or gained by playing it from Hidden, which gives Reaction while facedown. Satchel Charge is the Deploy card that already has Action built in.

Worth separating from the gear trigger: once a Deploy gear does enter, its "when you play this" ability does open a chain inside the showdown, and that chain can be responded to.