I'm statically linking Allegro with all library files included in the 1.5.0 release of the A5 dependencies which are:

libdumb.a
libFLAC.a
libfreetype.a
libjpeg.a
libogg.a
libopus.a
libopusfile.a
libphysfs.a
libpng16.a
libtheoradec.a
libvorbis.a
libvorbisfile.a
libzlib.a

and the main Allegro library of course

But I'm still getting undefined references. Here's one:

C:\Programming\Allegro5\lib\liballegro_monolith-debug-static.a(wsystem.c.obj)||In function `win_initialize':|

C:\dev\allegro_winpkg\universal\allegro\src\win\wsystem.c|196|undefined reference to `__imp_timeBeginPeriod'|

Maybe I'm missing some libraries?

You have to link to the windows libraries.

They are :

 -ldsound -lgdiplus -luuid -lkernel32 -lwinmm -lpsapi -lopengl32 -lglu32 -luser32 -lcomdlg32 -lgdi32 -lshell32 -lole32 -ladvapi32 -lws2_32 -lshlwapi -static-libstdc++ -static-libgcc

New errors now. Looks like something related to DirectX, which I have installed already.

..\..\..\Allegro5\lib\liballegro_monolith-debug-static.a(d3d_display_formats.cpp.obj)||In function `_al_d3d_generate_display_format_list':|

C:\dev\allegro_winpkg\universal\allegro\src\win\d3d_display_formats.cpp|49|undefined reference to `_Unwind_Resume'|

You're not using the same compiler that was used to build the dependencies. They have to match exactly.