This page offers a guide on how to accept and complete Request 1: Wurmple Can Evolve in Pokémon Legends: ARCEUS.
Where to Find a Wurmple in ARCEUS
Luckily for you, the next time you pop out into Obsidian Fieldlands, you’ll be able to find a group of Wurmples relatively easily near Horseshoe Plains, which is east of Aspiration Hill and the starting camp.
Wurmple can be caught either by sneaking up to them and throwing out a Pokéball, or you can engage it in battle by throwing out your chosen Pokémon. Sometimes Wurmples might pop out of the Pokéball if you’re trying to catch it outside of battle, and if they spot you they’ll run away (and disappear) - they’re not an aggressive Pokémon so they won’t attack you on sight.
Once you’ve caught one (or a few - afterall, you’ll want to build up your Pokédex!) and returned to Jubilife Village, remember to speak to Beauregard outside the Galaxy Team building. He’ll then ask if you have a Wurmple, to which you can then hand him over one. If you have multiple Wurmples, you may want to give him the weakest (he doesn’t care about levels) so you can keep the strongest one for yourself.
Wurmple Can Evolve Reward
Upon receiving a Wurmple, he’ll ask you for advice on which of the following names to choose: Beauticia, Beautifred, or Beaugene. Pick whichever is your favorite to finish up the request. You’ll get x3 Dazzling Honey for completing this request.