Important note: This issue doesn't appear to occur on the browser version, only the download.
Viewing post in Deltarune: Lightners Live Plus comments
Ah sorry, apparently Godot just doesn't work properly with switch pro controllers, which I can't really do anything about since that's just a problem with the game engine unfortunately. I think you should be able to make it work if you add it on steam, using the "Add a Non-Steam Game to my Library" option, and then playing it from there with Steam Input?