In marked contrast to conventional string theory, ambitwistor strings remain
solvable worldsheet theories when coupled to curved background fields. We use this fact to
consider the quantization of ambitwistor strings on plane wave metric and plane wave gauge
field backgrounds. In each case, the worldsheet model is anomaly free as a consequence
of the background satisfying the field equations. We derive vertex operators (in both
fixed and descended picture numbers) for gravitons and gluons on these backgrounds from
the worldsheet CFT, and study the 3-point functions of these vertex operators on the
Riemann sphere. These worldsheet correlation functions reproduce the known results for
3-point scattering amplitudes of gravitons and gluons in gravitational and gauge theoretic
plane wave backgrounds, respectively.