Burley push button wheels and axle? I'm looking into building my own trailer and I want something similar to Burley's push button wheel system. 

My question is, where do I obtain similar wheels/axles?

Here is the Travoy, one of their trailers that uses such a wheel:
https://www.amazon.com/Burley-Design-Travoy-Commuter-Trailer/dp/B0038LPO5Y



Replacement ones sold by Burley are pricey, anyone else have any luck looking for a similar product?
https://www.bikeparts.com/BPC417920/burley-replacement-wheel-16%22-alloy-push-button-axle
If what you really want is Burley hub/attachment guts, just ask around for someone getting rid of a Burley trailer with rotted out or missing fabric. They've sold a lot of trailers and many of them sadly get put outside to languish before eventually getting thrown away once they get nasty.

Alternatively, Ali Express has a lot of the approximate kind of stuff you're looking for. I don't know if you know the term "stub axle," but that is broadly what you're talking about, an wheel axle like a car or a wheelchair where the support is only on side.