Can I use a module of modarchive.org in my game/application/etc...?

The only way to be genuinely fair to the artist who's music you want to use is to contact them and get their permission. We can't grant permissions on their behalf.

The contact information isn't in the module, what can I do to contact the artist?

You can try using the modarchive.org forums, specifically the Wanted: forum.

I can't get in contact with an artist who's music I want to use, what now?

If your project is small and free (and is non commercial) then it's likely that just giving clear due credit to the artist in your production is enough. If your production is for commercial purposes then you are technically left with no other choice than to find a different module where the artist is contactable, or have someone compose a custom module for your project.