Definitely cup and cone bearings are the way to go for a soap box cart.
R500’s would be perfect, they’re cheap, roll well, there’s very little sealing from the elements to cause any drag, they’re semi aero and they’re stiff. Almost like it was designed for the purpose!
Tyre wise, rolling resistance is key. I’d suggest the fastest 25c tyres you can get your hands on for not too silly money. 25’s are advisable over 23’s in this case as there’s evidence that 25c is the optimum width when it comes to rolling resistance. Usually carry said items in stock if you need them…