For a system to track movie requests, it's actually really simple to do with a Google account.
Create a new form, write whatever question you want, and for question type change it to "Text" to let people write in their selections.
Boom. Google even lets you embed the form on a webpage so you can set it up right on this site.
And you can set it so the results get sent to a spreadsheet, which you can then sort in alphabetical order to see which ones show up the most.
You can also set up different forms with ready-made responses, so if you want people to vote on a pre-selected list of movies, you can do that too.