Is there a place to buy Oyster cards at Gatwick airport?
I can not find the information anywhere and since I will be travelling zone 1-4 during my stay I think it is the best option.
This site says that you can buy Visitor OysterCards at the GatwickExpress ticket office, and also pre-order them and have them delivered. The Gatwick Express site says that the ticket office at Gatwick is open 24 x 7.
You may not need an Oyster Card at all if your debit card does contactless payments.
Gatwick trains now accept contactless payments
This includes Gatwick Express.
You can also use contactless cards as an alternative to Oyster Cards on TFL, so your total fares are capped. That site has information about whether your contactless card will work.
According to this Tripadvisor post:
London Airports: Heathrow Underground station and Gatwick Express station sell Oyster
Also stated on Gatwick Express's site:
You can purchase a Visitor Oyster Card online or you can simply visit the ticket window when you arrive at Gatwick Airport Train Station.
Also from visitlondon:
You can also buy an Oyster Card at the Skybreak Service Centre in both terminals and at the Gatwick Airport train station.
All that's lacking is an actual confirmation that someone actually has bought an Oyster there...
The Visitor Centres at Gatwick Airport close at 16:00 daily, so if you arrive in the evening, you won't be able to buy an Oyster card at one of Gatwick's Visitor Centres. Additionally, there are no Oyster Ticket Stops, shops authorised to sell and top up Oyster cards, at Gatwick.
You may be able to buy a Visitor Oyster card in your home country and thereby already have it when you arrive in the UK.
That said, you don't necessarily need to have an Oyster card on arrival in Gatwick. You can pay for rail tickets with some other method, and buy an Oyster card when you reach central London. In fact, you might not need an Oyster card at all; see my other answer for all the places you can legitimately buy an Oyster card and alternative ways to pay for Underground travel.