Can I install an above-the-range microwave above an exhaust vent?
This is a common problem. So common that you can buy a microwave oven that has an exhaust fan on the bottom for use over a cooktop. Power and control of the fan is all part of the microwave. The typical term is &quot;over the range microwave&quot;. Many companies make them, such as this GE:

Key things to look for are:

Microwave oven capacity
Width - typical is 30&quot;
Speed (typically measured in CFM) of fan

and the tough one is to figure out if the exhaust vent location will work. Download the installation manual before you buy anything and see if everything will fit or what (if anything) you will need to change.
Update based on picture:
The picture shows an in-wall fan. I used to have one of those before I redid my kitchen many years ago. It looks like an original from the 1960s or earlier. This type of fan really does not do a very good job, because it is pulling air across instead of up. It is likely to pull more air from the rest of the kitchen than up from the cooktop. It might be effective if it is very high power, but I doubt that is the case.
Replacing the fan is a very good idea. But then you run into another problem. Even without including a microwave oven, a properly mounted exhaust fan would be relatively low over the cooktop. The cabinets above the cooktop should have been a bit shorter so that they could be mounted higher up on the wall to allow more room for a fan and (now) for a microwave oven.
What to do? When I redid my kitchen in a similar situation, I ended up, with the help of my electrician and his tools, knocking a new hole in the outer wall in the right place to install a new exhaust fan and I removed the old fan and covered up the hole. In your case, that will not be as easy a task. However, even without the fan problem (which is a real problem - you need a fan over a cooktop), there simply is not enough space to mount a microwave oven under that cabinet and still have a functional cooktop.