Does Eldritch Archer work with crossbows?

Yes. Crossbow is a trait. All Crossbows have the Bow weapon type and have the Crossbow trait.

Rules wise, it is, and the description of Eldritch Archer explicitly mentions using crossbows:

    Bolstering your athletic and martial abilities with mystical talents, you achieve rare heights with the bow or crossbow—transforming arrows or bolts into eldritch ammunition

Obviously this isn't rules text but the intention is clear that crossbows can be used with this archetype.

https://www.reddit.com/r/Pathfinder2e/comments/nz8fwu/eldritch_archer_crossbow/

Eldritch shot isn't really something you're expected to be doing every round. Something like- Round 1- Enchanting Arrow, Reload; Round 2- Eldritch Shot; Round 3- Reload, Phase Arrow, Reload; Round 4- Eldritch Shot

Is very workable, especially if you're using a caster base for your main class. But if you can use a shortbow or longbow instead of a crossbow, you should probably do that. Crossbows are for casters and martials with specific options that support their use. As simple weapons they're just not as good as their martial weapon counterparts.

For anyone curious you can get repeating hand crossbow proficiency with one of the following:

    The archer dedication + it's 6th level feat advanced bow training. You will still need GM approval for access to the uncommon weapon.

    The drow shootist dedication + it's 4th level feat repeating hand crossbow training. You need GM approval for the uncommon dedication, but otherwise good. This also gives access to the very useful associated bandolier.

    You may be able to use the human feat unconventional weaponry while having martial weapon proficiency, but you will need GM agreement that the repeating hand crossbow is common in drow culture. That might be a bit of a reach, but that would also solve access too.

    fighter + it's 6th level feat advanced weapon training. You still need GM approval for access to the uncommon weapon. Technically the feat isn't required either, but having an affective -2 to attack feels bad.

Anyways, you can't really get access to the repeating hand crossbow without GM approval. So, if you want to use one, make sure you run it by your GM. Also the feats dual thrower and dual-weapon reload in the dual-weapon warrior archetype also work really well with the weapon. Although that probably does not mix very well with the eldritch archer archetype.