Learn Polish: Chętnie angażuję się w każdą akcję. - I gladly get involved in every campaign.

The verb angażować się (to get involved) specifically governs the preposition w plus the Accusative case. Even though in English we say involved 'in', in Polish you should think of it as directing your effort 'into' a project, which implies the kind of metaphorical movement that triggers the Accusative.

Is akcja a false friend here? Does it mean the same thing as the English word 'action'?

It can mean physical action or a movie genre, but in everyday Polish, akcja very often refers to an organized campaign, a charity drive, or a specific social initiative. That is exactly what it means in this sentence.

Could I say lubię angażować się instead of using chętnie?

Yes, you can, and it would be perfectly understood. However, using the adverb chętnie (gladly or willingly) directly with the main verb is a very common and idiomatic Polish way to express that you are happy to do something, often sounding much more natural than using lubię (I like).

Why is the imperfective verb angażuję się used here instead of a perfective form?

The word każdą (every) is the big clue. Because the speaker is talking about a recurring habit—getting involved in every campaign that comes up—you must use the imperfective aspect to show that it is a repeated, ongoing pattern.