The principle of relativity is one of the most basic assumptions built in to standard physics. Fundamentally, it says that there's no single choice of a "correct" state of motion, which means that there cannot be any "experiment the bodies themselves can do to determine which one is moving" because there is no correct answer to that question — the only things that matter are relative velocities (because there is no such thing as "absolute" velocity). The issue of "which one is moving" depends on which frame you're in, and any frame is as correct as any other frame. So that's why the answer to the main question is "NO".
But there's another different question here, having to do with the Doppler effect in air, and it brings up an interesting subtlety. Only relative velocities matter, but now you have three things — the two bodies and the air — and the relative velocities between each pair of them will come into the physics. In principle, you could decide if one is at rest relative to the air. But there's nothing absolute about the air; it's just another object in physics. The wind could start to blow, and now suddenly the other object might be the one that's at rest relative to the air. Moreover, someone could drive by and decide that all three are moving with respect to her. But the principle of relativity says that she should be able to calculate the Doppler shift and — assuming she correctly accounts for the fact that the air is moving relative to her — get the right answer.