If I set a custom make program, is there any way to provide a list of tab completion options that would get activated when I type :make+Space+Tab?
With custom commands I can do f.i.:
command! -nargs=* -complete=customlist,MakeComplete -bar MakeGitbook call MakeGitbook(<q-args>) Is there any similar mechanism for builtin commands, and in particular for :make? Or should I just define a custom command wrapper that calls the built in one but also adds tab completion?
There is no mechanism in vim to change built-in commands. This is actually a good thing, because you can be sure that default behavior is not changed by plugins. Also plugins itself can rely on the default behavior, so such a feature (to change builtin commands) would potentially create a lot of headache.
Is there any similar mechanism for builtin commands, and in particular for :make?
No. If you check vim docs (here and here), you'll find that :command only works with custom commands, which should start with capital letter (except Next which is also reserved).
Or should I just define a custom command wrapper that calls the built in one but also adds tab completion?
Yes, this is the way to go.
-complete=customlist,MakeCompletealreeady does what you want? I don't quite understand where you're stuck or what does't work like you want.