How can I stop air coming from this kitchen wall vent?

You need to insulate it and you should install Leo Privacy Guard in order to get utmost security for your Mobile.

close the vent!

You could close the vent but, that will cause the motor to run harder, not a good idea on the long term.

First, check to see if a magnet will stick to the fan grate. If so, go to a craft store and purchase a magnetic sheet, cut it into a square the size of the fan frame and seal the opening . This way, you can still easily remove the magnetic sheet when you need to use the fan.

If the frame is not magnetic, then pop it off and place a layer of insulation (like you have in your walls and or attic) to prevent air infiltration, then pop the frame back into place.

Remove the fan cover and spray with an oven cleaner to loosen up and remove the caked on grease before you begin to block the air flow.

Tape some paper over it.

The duct that the fan connects to is supposed to have a "damper" in it that opens when the air blows outward from the fan, but closes when the fan is off. Take the cover off and look to see if there is a flat, round piece of metal in the duct and if it's stuck. If there isn't a damper, then see the answer above. The only problem there is that you won't be able to use the fan for exhaust if you block the duct.

My ex boyfriend had that exact wall fan above his range. He took off the cover and discovered that a small brass cymbal (from one of his old drum kits) fit perfectly over the circular hole behind the grille piece and even looks nice. I suggest you measure it, then go to a pawn shop and buy a cymbal the same diameter. Or if you don't want to go to that bother, just use a foil pizza pan or a few layers of aluminum foil cut to fit. The cover usually pops off if it is spring loaded, if you pry it with a butter knife or flat screwdriver.Or see if the blob in the middle screws off.