I live fairly close by and I've never seen any car hire facilities actually at the port. (Indeed, I've never seen much of anything there. There's the rail station and a basic café, and that's about it). Further, Stena Line's website makes no reference to car hire at Harwich.
The port itself is some way out from the town of Harwich. Even Harwich has hardly got any shops or facilities, as it's nowadays really little more than a village, right next door to the town of Dovercourt (which is where all of the shops, etc, are).
However a quick bit of Googling shows that Europcar operates a 'meet and greet' service at the port:
http://www.europcar.co.uk/car-hire-HARWICH_PORT_MEET__GREET.html
and Thrifty can provide you with a car from there (but it has to be returned to Colchester):
http://www.thrifty.co.uk/car-hire/harwich-international-port.htm
IndiGo also seems worth checking out:
http://www.indigocarhire.co.uk/uk-car-hire/harwich-international-port/
Chris