Meloetta can change forms during battle without needing a special Ability. Instead it uses its signature move, Relic Song. Cherrim changes form in the sunlight. I suppose Mew, Smeargle and Ditto (non-Imposter) can also change via Transform, but I don't know if that counts as a form change.
Primal Groudon and Kyogre change forms when they hold the Red an Blue Orbs, respectively, however like Rayquaza this could be counted as a Mega Evolution-esque transformation.
>What about mega ray? Form change technically, and it can be triggered by knowing dragon ascent
commented 18 minutes ago by Đarkheart
As for Pokémon that change with Abilities…
Zoroark changes appearance via Illusion.
Darmanitan changes form via Zen Mode.
Castform changes form via Forecast.
Ditto changes form (?) via Imposter.
Aegislash changes form via Stance Change / King's Shield.