Can I enable gifting of memberships?

Unfortunately, Shopify doesn't provide a great way to offer membership gifting via their Subscription API. This is because the membership subscription is owned by the customer who purchases it - there is no way to transfer ownership of a subscription to another customer (ex: a gift recipient).

The best way to offer membership gifting is via selling gift cards. Gift cards can be used by gift recipients to purchase a membership. This ensures that the membership is attached to the recipient (as opposed to the gift giver).