You would have to run this past your AHJ (Authority Having Jurisdiction). Your AHJ may want you to make the fireplace inoperable soas the next owner of the house, doesn't try to start a fire. You would need to line the chimney with metal ducting (10" round) and provide a termination cap. If you didn't line the chimney the grease would soak into the brick and leach thru the walls.

While it is possible, in order to do it properly may not be worth it.

The chimney may be used as a chase for the ductwork meaning, the pipe must be run up entirely through the chimney to the top with a proper vent roof cap. This can not always be done since the chimmney may not be big enough or takes a few turns on its way.

Before doing, a good inspection, perhaps with a camera would be required; the vent hood specs would need to be checked for pipe and max length allowed. Most importantly, the chimney must be checked to be sure it doesn't share common venting with any other appliances or wood/gas burning fireplaces.

If the chase turns out to be usable, then you must totally disable the existing fireplace to the point of bricking or sealing it off so that it is not used by mistake, including the unknowing next owners of your house. This alone may be the deal breaker as removing a fireplace may ruin the charm or appearance of your home.