Here in the UK coffee carts/trailers are a common sight and present problems for their owners with regard to power and water just the same as you will no doubt encounter.
Large espresso machines generally need a fair amount of power, 4500 watts or more for a typical 2 group and this usually means a large and expensive generator to tow around.
There are other options but be careful and give it plenty of thought before committing to buy.
First, consider a 2 group compact machine. Often 3kw or less, they need a smaller generator to power them. They will pretty much compete with a regular 2 group when making coffee and steaming milk but will lack the hot water output for teas etc. This makes them a good machine to pair up with a hot water boiler/urn that is gas powered.
Gas fuelled espresso machines are an option but in my experience they rarely have the output of their electrically powered equivalents. Regulations/certification for gas can be a hassle as well as carrying the heavy gas containers around. Bear in mind that gas powered machines normally still require an electrical connection for the controls and water pump to work.
If I remember correctly, Izzo, a company near Naples in Italy manufacture a gas fired lever group machine (no pump) with a 12V electronic filling mechanism but this is a rare thing...
Lever group machines do away with a lot of the electronics associated with modern machines and set-up correctly can make decent shots of espresso but still need power and or gas. Easy to service though and tend to be very reliable. They also dose out the correct size shots with one or two pumps of the lever leaving your hands free to start steaming milk or whatever?
Some machines here are powered by an 12V inverter and a bank of batteries, this can work quite well but the batteries have to be re-charged regularly and have a limited output/life. The more powerful the machine, the quicker the batteries run flat so this can be a better bet for gas powered machines that just require a small power supply for their controls.
We have sold many machines for outside use where power is a problem, generators can be the easiest option but you also need to consider where you are going to be sited. The drone of a generator can soon annoy your neighbours...
Look out for machines with "high groups" - more clearance under the groups for taller take-away cups so often used with carts.
I was in New York last week and I didn't see a trailer like that, but maybe I just missed it. I think it is a fabulous idea and would be a fun business venture. Keep us updated on your progress! How much coffee does a food trailer like that typically sell?