Its your ECM (ECU Same thing...), I'd put money on it.
The symptom of turning the key to on and waiting an unknown amount of time for the pump to growl and I bet your check engine light comes on at the same time
mine took upwords of twenty minutes some cold mornings back in 2009. Replaced the computer, no problems since. I went through autocomputerexchange out of FL. Its a pretty common issue on our 94-95 YJ's. You can replace the capacitors to if you want to rip into it. There are some threads on it out here.