Fred has the right idea.
A few more thoughts.
Because moped laws vary so much from state to state, contact people in each state you will pass through and make sure that you and your ped will be legal.
In Michigan, even the smallest two lane country roads have a 55 mph speed limit and a lot of them are dirt. This is not shown except on very detailed county maps and only one brand of those that I can find. Most state highways are also 55 but have a wide, paved shoulder which is great for riding a ped without blocking traffic. I am sure that people from other states can give you similar info.
Make a list of moped shops that carry your brand in case you need parts or other help. Even shops not on your route can fedex to you if they are the only ones with the part.
Ask for people who use the forum and are potentially on your route to email you with phone numbers in case you need help on the way.
You can most likely find places to stay this way too.
Carry a good tool kit, enough for roadside repairs and flat tire fixing and simple spare parts like spark plug, light bulbs, spare tube etc.
Maybe you can hook up with some other people on the way who are willing to ride too. A long distance swarm?
I hope you do it,