I am planning a bike to move heavy things, separate question here, and sixtyfootersdude is doing a cargo trailer that converges to some extent with my goal to move cargo here. I am not sure whether mounting a trailer wheel is the same as mounting a normal tire but bear with it I am trying to organize sixtyfootersdude's question at the same time as solving my own problem, just answer the question in the title thank you.
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lantius is almost right IMO.
You will need to use small wheels on a two wheel trailer (or build your own hubs) because the side forces in corners are more than a 26" or bigger wheel can cope with. With a solid build and care when cornering a 24" wheel can work, but a 16" or 20" wheel will just work. If you build your own hubs with 100mm or more between flanges you could use bigger wheels, but you end up with a very wide wheel (you just added 100mm to the width of the trailer without adding to the load space).
Finally, you must look at Richard Guy Briggs' organ trailer. It's a bike trailer that carries a Hammond organ and player.
The wheel is a pretty simple mechanism. You've got two main options:
1) Through-axle. In this case, the trailer has structural pieces on either side of the wheel. For example, this design is seen on the Bob Yak and Ibex and the Haulin' Colin trailers. The main advantage here is that you can use a normal bicycle hub and you have good structural support. The disadvantage is primarily that the wheel well increases the width of two-wheeled trailers.
2) Wheelchair hub. (See: Phil Wood) This is the design used by the Blue Sky Cycle Cart. A wheelchair hub has an axle you bolt down from only one side. This allows for a narrower body on a two-wheeled trailer, but you must design the axle fastening system carefully or risk having precession loosen it up. There is little reason to use this on a single-wheel trailer, unless for compactness when folding or using some kind of mono-suspension.
Both types of hubs can be built up with any rim. Smaller BMX-sized wheels (ERTRO 406mm) have the advantage of taking up less space, light weight, and strength, but the normal disadvantages of smaller wheels when it comes to rolling over obstacles. 24" (520mm) and 26" (559mm) wheels are larger, but you should design your trailer bed to hang below the axle for maximum stability. Larger wheels are typically only suitable for special applications or in an Extrawheel-type design.
You can now buy the Surly trailer wheels. They are beautiful! The hubs are only about 35mm wide! Retail for $170 per wheel.