No, you don't need a computer at all. The Netatmo just transfers data to your "station" via WIFI. So, as long as it is connected to your wifi network and maintains an internet connection it will transmit the data.
Couple other notes, according to their tech support:
1) If you lose internet connection the base station saves the data and will transmit the information once your internet connection is restored.
2) If you lose power to the base station you will lose data until power is restored.
I really wish they had a built in battery backup for this, as I don't want to lose precipitation data from my rain gauge during storms, which is when it is most likely to lose your power. As a result though, I am contemplating buying a UPS to try to prevent data loss during power outages.