The acronym NEET is a rather specialised word, whose meaning is changing.
It was originally used by the government in the UK as a way of measuring school performance. A good school would get its students into university, job or training after they finish, so the government measured the number of people who were Not in Education Employment or Training 6 months after leaving school.
It was originally used like an adjective:
The school has a low number of people who are NEET at six months.
It then became adapted to be used as a noun.
The school has a low number of NEETs at six months.
It then picked up a perjoritive sense, especially in Asia. It shifted in meaning from "people who were unable to find a job" to "people who chose not to look for a job". This is the sense that you have learnt. But be careful, since this meaning of NEET is not widely used outside of Asia. In Britain NEET is still mostly a technical word used by school managers. I don't think the word is used much in American English at all.
I'm worried that Hitoshi will become a NEET. All he does is sit in his room and play video games.
Now, the question that you ask "what is verb for NEET" doesn't make much sense. For example, what is the verb for "apple" or the verb for "secretary". In general there isn't a way to change a noun "X" to a verb that means "being an X".