You are comfortable running your cart because you are on your own playing field. When you visit other businesses, you are on their playing field. Just remember that some folks can be jerks and don’t take it personally. Just ask to see the owner/manager and introduce yourself by saying “Hello, I am xxxxx from xxxxx catering – we own a hot dog cart and thought it would be good for both you and I if I could get permission to operate my cart outside of your establishment. blah blah blah yada yada…. It’s easy once you break the ice.
Also remember that it is usually just a matter of time before people will start asking YOU about doing other gigs!