Can I sell gift cards?

Yes, with the Gift Cards plugin. There are two ways:

1. Issue cards from the admin, for example as a goodwill gift.
2. Sell gift cards as products: on any product, tick This product is a gift card and set the amount (a fixed price, or a minimum and maximum so the buyer chooses).

When the order is paid, a card is issued and emailed to the recipient with the buyer's message. The recipient redeems it into their wallet as store credit, so you also need the Wallet / Store Credit plugin.