Do I have the possibility and the right to connect a personal card issued by an Israeli bank to a nonprofit business account of a nonprofit registered in US?
I would say no. Well, .. I voted no. I would think personal would have to be kept separate from business unless there are some specific terms with your card that offer you to make the connection.
It seems incredibly unprofessional. Get the non-profit to get their own card and add it. If they have any credibility that shouldn't be a problem but from your description, I'm wondering if credibility might be an issue?