How do I check if a Nintendo Switch game card supports multiple languages?

Check the Nintendo eShop, by searching for the game title. In the details (no. players etc.), it should say supported languages - this should work since most if not all physical releases should be on eShop

It depends.

If the game supports any other language than Japanese, you may play it in that language. It is not given that a game or a software will support languages of other regions, though. For example if you buy a game from Japan, you can't be sure it supports English. You should ask the vendor or read from the box what languages the game supports.

Your Nintendo Switch system tries to launch the game in your system's default language.

Source: https://en-americas-support.nintendo.com/app/answers/detail/a_id/22354/~/can-i-play-my-game-in-another-language%3F