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The operation of free-electron lasers (FEL's) with axial electron beams and high-power electromagnetic wiggler fields such as those produced by high-power gyrotrons is discussed. The use of short wavelength electromagnetic wigglers in waveguides and resonant cavities can significantly reduce required electron beam voltages, resulting in compact FEL's. Gain calculations in the low- and high-gain Compton regime are presented, including the effects of emittance, transverse wiggler gradient, and electron temperature. Optimized scaling laws for the FEL gain and the required electromagnetic wiggler field power are discussed. Several possible configurations for FEL's with electro-magnetic wigglers powered by millimeter wavelength gyrotrons are presented. Gyrotron powered wigglers appear promising for operation of compact FEL's in the infrared regime using moderate energy (<10 MeV) electron beams.