First look for the presence of a steering damper between the steering linkage and cross member. It will look like a shock absorber but it pulls apart and pushes together equally hard in either direction. If there is one, check if oil has leaked out where the shaft comes out. If so, replace it.
If there is no factory-installed damper, adding one might solve the shimmy but that should not be necessary to fix it now if it wasn't needed before. Inspect the idler arm first, then the tie rod ends and ball joints. If you don't know how to inspect them for play and wear, have that done at any tire and alignment shop.