No. You need a machine- or office-issued ticket on orange and yellowy-green card.

The ticket is an 'Anytime Return', and costs £31/person for Southern, or £34.90/person for Gatwick Express (also valid on Southern).

If you're going for this ticket, there's no price advantage to prepurchasing online. There's nothing to stop you from doing so, but you'd still need to pick it up from a ticket machine at Gatwick Airport, using the credit card you paid with. This ticket can be used on any train - there are no seat reservations on these services anyway.

You can buy the same two ticket types from machines or ticket offices (or mobile ticket sellers) at the airport. Bear in mind that, if you're wanting to go for the Southern ticket, these are heavily skewed towards getting you to buy the more expensive Gatwick Express ticket (although the difference is relatively small in your case - for other journeys it's much bigger!)

Advance Purchase tickets that are restricted to a single train are sold on a one-way basis only (so a round-trip would be two one-ways). This means that the 2-for-1 offer is only valid on the date on the ticket, and you'd need a paper travelcard to cover you for other days.