In the working timetable, the Milford Haven train is booked to arrive at 06:17½, and the Fishguard Harbour to depart at 06:20 (which in practice means, if it's running on time, the doors will be closing around 06:19½). So if you're feeling lucky and think you can make the dash between the two platforms (which looks to involve a long ramp, around the middle of the arrival platform, up to the road overbridge and back down again), I suspect a young(ish), fit, and healthy person could make it in favourable conditions. If anything's running slightly late though, you'll likely be waiting an hour for the 07:20.
EDIT: Ah, I forgot it was a request stop. Yes, this could make it harder, as you'll want to be on the platform by the time the train is arriving so you can give a hand signal. Possibly still doable. I wouldn't make plans that revolve around me being able to do this connection!