Quick Answer
Can 'Em All doesn't list controller support on Steam. On October 31 the developer said full support wasn't in yet and was planned for the next patch. We haven't seen a patch note confirming it, so plan on mouse and keyboard for now.
Does Steam list controller support?
No. Steam lists Single-player, Steam Achievements and Family Sharing, with no Full or Partial Controller Support label.
That doesn't mean your controller can't work at all. See the Can 'Em All Steam overview for the rest of the store details.
What did the developer say?
The developer said full controller support wasn't available yet and planned to add it to the full game in the next patch. Check the patch notes on Steam before you rely on it.
Can you use a controller anyway?
Player reports are mixed: one player said they played the Can 'Em All demo with a controller, and another couldn't get theirs working. Neither shared a controller model or Steam Input setup, so there's no setup to copy yet.
If you want to play on a handheld, read Can 'Em All Steam Deck. If you mean a console, see Can 'Em All Xbox and PS5.
FAQ
Not as far as we can confirm. The developer planned it for a patch after October 31.
One player says yes, another couldn't get it working, so try it before counting on it.