Yes, Firestick works with satellite Internet. The real question though is which Internet Service Provider (ISP) is being used and what the data cap is of that plan. Firestick is a device that allows a user to run applications that stream movies, tv shows, videos and other entertainment options. Netflix, Hulu and SlingTV are examples of these. Netflix recommends a user having a download speed of 25Mbps to stream 4K video. If that isn't an option then streaming in high-definition video and in standard definition requires 5-10Mbps and 3Mbps respectfully. This is where it is important what the data cap is, because if it is one of the lower options then it won't take long to reach the data cap streaming through the Firestick everyday. If the data cap has been exceeded then it will usually result in slower performance or an additional charge depending on the users ISP. These data caps are used by the ISP to balance large amounts of traffic on the network. Viasat has higher data caps compared to other ISP’s so if exceeding it becomes a problem then there are other plans to choose from that may work better.
It is important to keep in mind that the more devices that are in use at the same time, the higher the download speed needs to be to accommodate them all. If the devices are running slower than normal, it would be good to check which devices are on and being used at the time. It is also good to check the internet speed on the Firestick to see where the issue is coming from.