Can Gust be played with no legal target?

No, you cannot play Gust if there are no legal targets on the battlefield.

According to RiftJudge FAQ #1294, you cannot play any spell that does not have a legal target to begin with. Furthermore, Core Rule 359.3.e.7 states that if all of a spell's targets become invalid by the time it is finished being played, the instruction will not execute, but the spell must have had a legal target at the moment it was played to be valid.

If you are looking to play Gust to trigger an effect that cares about spells being played (even if the Gust itself ends up having no effect), you must still select a valid target at the time of playing it.