Can you play Solari Shieldbearer with no other units in play?

Yes, you can play Solari Shieldbearer with no other units in play.

According to Rule 055, when executing card text, you do as much as you can and ignore impossible instructions. Because the ability "stun a unit" does not require you to have another unit already in play to play the Solari Shieldbearer itself, you can successfully play it. If there are no other units on the board to target with the stun ability, that part of the effect will simply not be executed (as described in Rule 359.3.e.6).

Note that per RiftJudge FAQ #7116, Solari Shieldbearer is a valid target for its own ability if no other units are available.