I was wondering if there was a way to cross compile an allegro program for Windows using MinGW on Linux to run on a Windows PC?

You an look how Allegro itself is cross-compiled: https://github.com/liballeg/allegro5/blob/master/cmake/Toolchain-mingw.cmake

Basically tell cmake which compiler to use instead of the native one, and it will compile for Windows instead of Linux. Getting all the dependency libraries is still very involved unfortunately, as is getting the resulting .exe to actually work in a native Windows (needs to find all the right DLLs and stuff).