The only character my opponent has in play has Ward. Can I still play Last-Ditch Effort and give my character Challenger +2 even though I can’t choose an opposing character?
Yes. When there aren’t any eligible opposing characters to choose for the first part of Last-Ditch Effort, that part of its effect simply finishes without exerting a character. However, the second part of its effect still happens, so you still choose a character of yours to get Challenger +2 until the end of the turn.
Official set release notes — Fabled