Hey Jenna,
Thanks for reaching out!
While we could do something like this if necessary, it is actually not necessary.
Licenses do not affect plugin functionality in any way, shape, or form. You do NOT need the license key to install the plugins, use the plugins, etc. License keys only sign up a domain name for support, and for automatic updates.
So, if you are moving your site to a new server, simply go into the existing site location’s /wp-admin and to Events → Settings → Licenses. Remove the license key. Then go to http://theeventscalendar.com/license-keys and “disconnect” the license key from the old domain name.
Then, move your site.
On your new site location, add the license key in wp-admin, and you’re good to go! 😀
I hope this information helps.
— George