Is it wrong to have a relationship with someone at work that is not an empoyee but a vendor?

Possibly, it depends on what you do and what the other person does.

For example you cannot be a purchasing agent and have a personal relationship with someone at a another company that represents that commpany or is in management of the company that you are purchasing goods or services from.

I don't think it is wrong but the big thing is no PDA at work this will make you look trashy and people will lose respect for you or gossip about you... and I am sure you do not want that.