  It has been shown previously that a large class of monomial maps equivariant
under the action of an infinite symmetric group have finitely generated kernels
up to the symmetric action. We prove that these symmetric toric ideals also
have finite Gr\"obner bases up to symmetry for certain monomial orders. An
algorithm is presented for computing equivariant Gr\"obner bases that
terminates whenever a finite basis exists, improving on previous algorithms
that only guaranteed termination in rings Noetherian up to symmetry. This
algorithm can be used to compute equivariant Gr\"obner bases of the above toric
ideals, given the monomial map.
