Can you use Sprite's Call on a Battlefield when someone else starts a Showdown and you have no units?

No, you cannot use Sprite Call in that situation.

According to RiftJudge FAQ #3336 and RiftJudge FAQ #2274, you can only play units to your base or to a battlefield that you currently control.

If an opponent moves into an uncontested battlefield (or one where you have no units), you do not control it. Because you have no units at that battlefield to establish control, it is not a valid location for you to play a unit, even during a showdown.

As established by Rule 3310, while you can play units to an occupied battlefield during a showdown if an effect allows it, this requires you to already have a defending unit there to maintain control of that battlefield.