Need a program to collect cache of join mailing list emails into a single document.

General information for the website: fashion week new york company
Kind of development: Customization of existing website
Num. of web pages/modules: 1
Description of requirements/features: Basically we have a place on our website where people enter their emails to be added to a join mailing list. This form only sends the person's email address to our join mailing list email with the subject "join mailing list request" and someone from our team has to manually add them to a word document where they are all collected. it takes a very long time and we would like these emails to be automatically added to a document (perhaps a google drive doc?) automatically, so that a human doesn't need to do this.
Extra notes:

I made our website and I have absolutely no idea how this would even go about being done let alone what language it would be written in. Any suggestions or solutions on this matter would be greatly appreciated, even f you don't know how to write the program yourself. Any tips in the right direction will be helpful. Thanks!

WEBSITE: http://newyork-fashionweek.com/