You could either run it all direct from a PC, or use a micro-controller to read the sensors, and transfer the data to a PC - it really makes little difference.
As you're only talking about less than 20 sensors it's fairly simple to do.
My current PC based wireless monitoring I'm playing with would easily do 20 'as it is', as it currently uses 1 byte ASCII codes to identify each sensor (I'm using A, B, C and D for four).
Or as already suggested, a PI would easily do it - or an Arduino, or PIC based board.