What would the products be for a reaction with ammonium nitrate and water as the reactants?

There is no reaction -- at least none that would make sense to write down. Ammonium nitrate does dissociate:

NH_4NO_3 (aq)\, \rightleftharpoons \,NH_4^+\, + \, NO_3^-

One interesting thing about mixing ammonium nitrate and water is that there is a large heat of solution involved. The mixture gets very cold.

NH_4NO_3 + heat \rightarrow NH_4NO_3(aq)