Thanks for all the feedback; here are my notes for possible causes so far:
No steering damper
Alignment (@gnob)
Caster too low (@gnob, @gummycarbs, @toyotaboy80, @baldilocks)
Unbalanced wheels/tires (@jpoole, @MrMacdugal, @flyingmonkey)
Panhard bushings (@AussieHJCruza, @Squeegee, @mudgudgeon...