It makes perfectly good sense that it only leaks when operating in the rain because the vents or louvers are on the side of the unit. So you pretty much need to be sucking the rain in horizontally.
What happens if you just turn a hose on it with unit off? If it leaks now without the unit running, then the roof seal is the most likely culprit.
I assume the water is coming in where the vent/filters are located, often in the kitchen area.
If there is not an obvious roof seal leak, I would just take the outer cover off take a look.
If the plastic inside cover is cracked or deteriorated with age and heat from the sun, you should see something out of the ordinary.
If the integrity of the inside cover has not been compromised, take that inside cover off as well, and see what you can see. If it is a seal it seems like you should see some pattern of water stains inside.
Heat can be a real problem. The plastic squirrel cage blower wheel on my original Carrier A/C unit self destructed about year 4. No leaks, but lots of noise and plastic shrapnel. So anything is possible given how hot things get on the roof.
