  Pulsar timing arrays are sensitive to gravitational wave perturbations
produced by individual supermassive black hole binaries during their early
inspiral phase. Modified gravity theories allow for the emission of
gravitational dipole radiation, which is enhanced relative to the quadrupole
contribution for low orbital velocities, making the early inspiral an ideal
regime to test for the presence of modified gravity effects. Using a
theory-agnostic description of modified gravity theories based on the
parametrized post-Einsteinian framework, we explore the possibility of
detecting deviations from General Relativity using simulated pulsar timing
array data, and provide forecasts for the constraints that can be achieved. We
generalize the {\tt enterprise} pulsar timing software to account for possible
additional polarization states and modifications to the phase evolution, and
study how accurately the parameters of simulated signals can be recovered. We
find that while a pure dipole model can partially recover a pure quadrupole
signal, there is little possibility for confusion when the full model with all
polarization states is used. With no signal present, and using noise levels
comparable to those seen in contemporary arrays, we produce forecasts for the
upper limits that can be placed on the amplitudes of alternative polarization
modes as a function of the sky location of the source.
