Water leaking onto stove from overhead exhaust fan during rains.

We had some water leaking onto the kitchen stove from the exhaust fan above it during the last couple of heavy rains. During the last storm it even started leaking through the ceiling light which is about 4 feet away from the exhaust fan on the ceiling.

This is no fancy over the stove exhaust system just a round circular fan that sits flush up against the ceiling and vents to the roof. I can't get up to the roof to see what it looks like from up there due to a recent injury but I'm guessing that maybe the flap isn't closing properly. I thought that maybe it was the seal around the the exhaust on the roof but I don't see any water in the attic behind the walls so I'm assuming the water is just coming in down the vent and some is leaking onto the stove. Does that sound about right?

Can't fix this myself. I'm wondering who fixes this kind of thing. Would it be a roofer or an appliance guy? If you say HVAC, we just had furnace replaced, should I call them? Thanks