HICan an extremely leaky valve cover (the oil is oozing out at a significant rate) cause a CEL to come on?

Nope, unless it seeped into an electrical connector, but even that is doubtful. Once the Check Engine light turned on, there should be a stored diagnostic fault code in the Engine Computer's memory. Many auto parts stores will read those codes for you for free. That will indicate the circuit or system that needs further diagnosis.

Don't confuse that with the "Check Gauges" light, if your car has one. That could be telling you the oil pressure is low which would be typical when the oil level is low.