Solids generally react only very slowly with each other, unless you put some energy into the system (e.g. by grinding them together) or the interface between them melts or liquefies somehow.

In the case of a frozen block of acid in contact with a piece of acid-reactive substance, there would likely be just enough melting or water vapour condensation at the surface to allow a reaction to occur -- this might in turn generate heat, that would keep the reaction going.