Learn Finnish: He haluavat mennä naimisiin kirkossa. - They want to get married in a church.

This is a core feature of how Finnish handles abstract changes of state. Just like physical movement to a destination uses the illative (the "into" case), entering a new condition or state often uses it too. The phrase mennä naimisiin literally translates to something like "to go into marriages", functioning as a fixed idiom for "to get married".

What kind of word is naimisiin originally?

It is the illative plural form of the noun naiminen (the act of marrying), which derives from the verb naida (to marry). While you won't see naiminen used as a regular noun very often today, its illative plural form got completely frozen into this everyday idiom.

Why doesn't the verb mennä make the church take the illative case (kirkkoon)?

Because in this sentence, mennä is not expressing physical travel; it's entirely tied up in the idiom mennä naimisiin (to get married). The church is simply the static location where the wedding event happens, so it takes the inessive case (kirkossa, "in the church"). If you said He haluavat mennä kirkkoon naimisiin, it would mean they are currently elsewhere and want to travel to the church in order to get married there.

Could I just use the single verb naida instead of mennä naimisiin here?