In the past this question would have had a simple answer, there is no Internet, but that has changed. Nowadays many safari lodges and camps have even Wi-Fi, but there are still camps that want to deliver the ultimate African safari experience without cell phone signal and without Internet.
Internet in safari camps and lodges is satellite based Internet and bandwidth is usually not very good. Recon that it is pretty slow when you try to use it at times when everybody is back from the game drive and wants to check the emails.
Some camps also charge for the Wi-Fi, often in packages of e.g. 30 minutes. They can be quickly gone when you have the bad luck of Internet rush hour and it is very slow. All he 5 star accommodations usually do not charge and offer Wi-Fi even in the room. Tented camps have the wireless usually in their main area.
Often cell phone signal is weak or instable. You can switch on your phone while you are on game drive (but set it on silent!) and while driving around there will be areas with good signal to fetch your messages and you can read them back in the camp.
Ask when you book your safari, if the lodge or camp has Internet, if you rely on it. Just to make sure you make the right choice.