Using a single long axle with bicycle wheels is a problem. the axle has to be skinny to fit inside the hub shells and most hubs use 3/8 or 10mm axles. That is just too skinny and weak to support the weight without bending. Also, the standard threaded rod thread pitch won't match up to bicycle bearing cones. You would have to buy solid rod and tap it to match.
Other alternatives:
There are some super strong bmx wheels for freestyle stunts that use 14mm axles but that would blow any budget.
Build a double frame on each side so both ends of an existing wheel axle get supported. I'd use bicycle front wheels as the flanges are wider and will have more lateral stability.
Use "cart" wheels that are designed for 1/2" or 5/8" axles. Those will have the bearings included. Available in many sizes and some even have inflatable tires.
There are one sided wheels used on many kid trailers that fold up for storing without tools. They have the brackets and have a quick release device on the hubs. Most are 16 or 20". Avoid the plastic wheels, those can't take much air pressure or weight. The wire spoke wheels are much stronger. Plenty of beat up kid trailers around that would be a good source for the parts.