I have the latest version of MAME, recently I attempted to run, Ninja Gaiden (arcade), but the emulator states that "the game is missing one or more of the required ROM or CHD images".

What can I do to fix this problem?

You could try testing it with an older version of MAME/MAME32. I would guess you have a bad romset but I'm assuming you already checked to see if this was the case. Run it with an older version and if you still get an error it is most likely related to the rom and I would recommend downloading it from somewhere else. You could also try downloading all the rom sets for that game to see if any of them work.

It might be possible that you need to download the bios pack for that rom too (like downloading the neo geo bios pack to run neo geo games).

If it works with the older MAME version then its either got something to do with your current MAME settings or theres still some kinks to be worked out...

You don't need a BIOS to run Ninja Gaiden Arcade. Your romset is outdated, or you are missing the parent set. Note that the US set is clone of the World set 1 named Shadow Warriors. Said set also went through rom name changes in 0.108u1.