Colemak is my main keyboard layout, so my vim installation is set up to automatically source a Colemak keyboard mapping file system-wide.

When I try to use vim to type in Arabic (via set keymap=arabic_utf-8 or set arabic), I find that the keymapping is off: the letters are Arabic letters, but the keymapping for input still corresponds to Colemak. So the Arabic letters are scrambled in a way that corresponds to how letters in the QWERTY layout are scrambled relative to the Colemak layout.

Is there a way I can change the language AND the keymapping to a standard Arabic layout for input mode while using the Colemak layout for Command mode?

Interesting question. I guess it is about two different layers, the Colemak keyboard mapping is a relatively low level access to the keyboard to replace all the events (= reorder the keys).
The arabic font is set on another level, where actually the key events will be translated to chars.
So probably you need to switch of that Colemak mapping (or revert) or find a combined arabic keyset for colemak mappings. Probably it can be generated too.