Learn Finnish: Me sotkeuduimme vahingossa naapureiden kiistaan. - We accidentally got mixed up in the neighbors' dispute.

This is due to the verb sotkeutua (to get mixed up in, to entangle oneself). Like many Finnish verbs expressing participation, interference, or moving into a situation (such as osallistua or puuttua), it requires the illative case. Conceptually, you are getting tangled into the dispute.

How does the verb sotkeutua relate to the basic verb sotkea?

Sotkea is a verb meaning "to mess up" or "to dirty" something. The -utua ending in sotkeutua makes it reflexive. It changes the meaning to "to get messed up" or "to get entangled," meaning the subject itself ends up in a messy situation rather than doing it to something else.

The word for "accidentally" here is vahingossa. What is the literal translation of this form?

It is the inessive ("in") case of the noun vahinko (accident, damage, or mistake). So literally, it means "in an accident." It is the standard, idiomatic way to say "accidentally" or "by mistake" in Finnish.

Could I use naapurien instead of naapureiden for "neighbors'"?