Easiest way to get hooked up with a P-3 fam flight would be to call the duty office, ask to speak with either the Squadron Duty Officer (SDO) or the Assistant Operations Officer (usually a LT, not as busy as the OPSO, who will be an O-4, with his hair on fire from stress). Don't call VP-30, they're the FRS, and won't have much time to spare. When you do get a hold of a squadron, try to find the one just back off deployment, they will be the least busy. Make sure you get and leave good contact names and numbers, and be persistant. 99% of squadrons will hook you up, just be prepared to wait a while, anywhere from weeks to months. I'd also inquire about the possibility of getting a little sim time, since Jax has at least 4 (i dunno, its been a looong time) full motion "pain machines" (painful for us, not you).
When you finally do go up, bring a water bottle and a snack. No number 2's in the onboard head unless you wanna clean it out afterwards.
I'd also bet the 60 guys in Mayport would hook you up as well. S-3s in Jax...no clue...couldn't hurt to ask, I'm not sure what their survival training requirements are for fams...