I am looking to get mod_rewrite to operate on the client's website who was adamant that people use Godaddy website hosting. She's got a shared linux hosting account. Will mod_rewrite work (using .htaccess) and when so, how? I am unsure if Godaddy uses Apache or otherwise.

Yes, GoDaddy's Linux hosting uses Apache, and mod_rewrite via .htaccess is open to you. A fast Google revealed numerous caveats you might encounter, top among them being 15-60 minute delays before .htaccess changes work.

Frankly, I avoid GoDaddy such as the plague, specifically for hosting. Only bad encounters together.