If Elder Dragon targets my unit with the on play effect, can I move that unit using Wide the Wind to avoid taking the damage?

I think you mean Ride the Wind — but unfortunately, no, you cannot use it to dodge Elder Dragon's damage.

Here's why: Elder Dragon's "When you play me" trigger goes on the Chain in Closed State. During Closed State, only [Reaction] cards can be played. Ride the Wind is an [Action] spell — it can only be played on your turn in Open State or during Showdowns. So you can't respond to the trigger with it.

What would work: Flash is a [Reaction] spell that does the same thing — "Move up to 2 friendly units to base." You can play it in response to Elder Dragon's trigger, and per the FAQ, units that move to a different location than where they were targeted will be unaffected by the damage. Tactical Retreat (also [Reaction]) lets you replace the death with a recall instead, which also saves the unit.