No, a computer is not necessarily needed to send your AIS data to FleetMon. Only if you want to use special software you need a PC.
An AIS receiver without a connection to a computer and special software is even better because you don't have to keep your PC running. A second advantage: you can place your receiver much closer to the antenna (but still indoors!).
There are different options to replace the computer, i.e. you use an AIS receiver with a built-in Ethernet port, you can use an AIS receiver connected with a serial-to-IP-converter (like the one you will get when you receive your station from FleetMon) or you could use a Raspberry Pi.