Drone regulation is still fairly loose compared to South Africa, but drones do have to be registered with SWACAA.
You should visit the nearest police station to notify them of the planned drone flight and purpose, prior to flying.
No licence or permit is required to fly commercial thus far, besides the basic registration with SWACAA, and there do not seem to be any regulations against foreigners flying drones in Swaziland.
Be aware that you may not fly over the airport or the Royal Residences. If you wish to capture a national traditional event such as the Reed Dance or Incwala Festival, a permit needs to be sought from the Ministry of ICT.