After you select your options and choose “this is a gift”, you enter your email address. Then if you click “subscribe now”, it is supposed to take that email across to the cart, however it doesn’t.
What I have found out is that it is not included in the “add to cart” Ajax request. As far as I understand it, the theme is bypassing the default “add to cart” process, to perform this ajax call instead. I assume that the theme hasn’t been tested with WooCommerce Subscriptions Gifting plugin and therefore, this recipient field is missing.