Hi Jeff Cotton,
I am really sorry for the delay of this reply as it is a weekend for us here.
I have checked in your site, it seems like you have disabled this user - Recaptcha apps from your backend Easysocial application page.
By the way, I already help you enabled it and assign this Recaptcha custom field in your "Registered user" profile type, so it will appear Recaptcha in your registration form now.
However I'm not very sure why you want to disabled with your registration option from backend. To test this, enable back with your "Allow user sign up" then try a new user registration in your site, you should see the Recaptcha field now.
Hope this help.