I'd make a control layer with a slider on it. Create one logo use the wiggle expression on it's position, but with 0 wiggles per second and the amount linked to the slider. Duplicate that a bunch of times, alt-drag new things into replace the duplicated ones to get a variety of logos. Then put them all where you want and keyframe the slider.
You could do it with Element 3D if you've got that. Put all the logos on different planes (simple object) as a texture in the same group, use the logo as the path for the emitter and animate the random displacement.