Interpreting "moving parts" as meaning that every solid part of the device has to be rigid, and the requirement as the capability to deliver a constant stream of air at a non-trivially greater-than-ambient constant pressure, I suspect the answer is a qualified no.
I'm also assuming that fuels and working fluids don't count as "parts."
It seems reasonable to assume that any remotely practical air compressor will operate under conditions where the ideal gas law applies, so $p \propto \frac{NT}{V}$.
One approach would be to imitate a typical air compressor and try eliminating as many moving parts as possible. For example, something like a hydraulic ram could eliminate pistons, impellers, screws, etc. and allow us to extract energy from a moving body of water to compress air, but it still requires valves. A valveless pump as seen in this video requires a special rotating piston. A basic siphon has no moving parts at all and it can create pressure if you enclose the lower reservoir, but is totally impractical as part of an air compressor—and even if it weren't, you'd still need some sort of valve to deliver the pressurized air.
Another approach is to manipulate temperature, which sounds like what you've got in mind. It's easy enough to generate heat without moving parts; a burner or an electric coil will do it. But how do you get around the valve issue? In order for the pressure to build, you need an enclosed space, and once you have pressure, the air needs to exit that enclosed space. If you wanted to get creative, you could try something like a diaphragm with an aperture that only opens when the diaphragm has expanded; the pressure will then make its own exit. But an expanding and contracting solid diaphragm or bladder also seems like a moving part, to me. It might be more durable than other types of moving part, I suppose, but then again it might not.
To produce a constant stream of pressure then you need a holding tank and the magnitude of your delivery pressure will be reduced significantly based on the upper limit of pressure you can develop in the holding tank and how quickly you can develop it. The Tesla valves suggested in netduke's answer are very clever but they're really differential flow-limiting devices; I don't see them being able to develop and hold pressure in a tank that you could release on demand for pneumatic power.
So the reason it's a "qualified" no is this. In theory, if you accept that your air compressor may be totally impractical for most purposes, you don't count elastic deformation as movement and you "cheat" a few times with valves and regulators, then yeah. You can create a device that compresses air in a tank, and then do with it what you will. In practice, it sounds like a poor idea that doesn't scale well, but it's an interesting exercise to play with.
Another qualification is that you might get a completely different answer in a microfluidics context.
