Diocesan priests don’t, but Fr. Corapi is a member of an order called the Society of Our Lady of the Trinity. I don’t know what their rules are in regard to poverty.
Their website says they take vows of poverty.
http://www.societyofourlady.net/brothers-formation.html
I saw Corapi a few years back; he was attended to by an attractive young woman, his “niece.” I said to my husband than, “funny, she didn’t act like his niece, and he was very wordly in everyway, even when he spoke. Not like the Father Corapi of EWTN.” I told my husband I was very suspicious, and felt Corapi had lost his way. And he seemed to live in a remote area under know formality. He seemed very wordly, as I said.