Can a microwave be exhausted through an attic vent?

Yes you can do that. If there is duct work existing you can reroute it to an existing attic vent. You might need to add an additional duct fan if the existing fan doesn't have enough force for the duct extension.

If this is a stand alone microwave, they require 3 to 4 inches of venting space on the sides, top and back and vent outward usually through the bottom. You could enclose the entire microwave and install a duct through the ceiling and into an attic vent but you'd also need a duct fan because the microwave fan is a circulating fan and not a directional one and you'd have to get the hot air out of the enclosed area.

If you tie it to an existing vent be aware of what that vent's for and which way it slopes. For example, a bathroom or dryer vent could be putting out lots of moisture that could condense inside the duct and run back down into the microwave. A gas furnace or water heater could do the same, just with less moisture. Another risk is the microwave/vent hood fan pressurizing and backdrafting the existing vent. If the existing is for a combustion appliance (eg. furnace, water heater, gas fireplace, or wood stove), a backdraft could be very bad. Another risk for a vent hood is grease buildup in the duct over time. If there's a fault or fire in a combustion appliance it could melt and/or ignite the grease, turning the duct into a veritable chimney fire.

I have my gas furnace and water heater both vented through my chimney, with lots of opening and draft to spare. When I put in a combo microwave/vent hood with a good sized fan (400 cfm? not sure), I asked the installer and local codes enforcement about venting the combo unit there as well. Both insisted on running a separate 4" minimum duct through the roof, citing a primary fear of backdraft and secondary risk of grease fire. You may want to ask your local codes enforcement to be sure.

One advantage to running the vent straight up through the roof is it gives you the shortest run and best draw to eliminate cooking heat, fumes, and moisture. My vent hood moves enough air on high that I can run the oven at 500 degrees, with 3 stock pots boiling, a large skillet frying, and my kitchen barely gets warm. I think a longer run with a couple elbows might have noticeably reduced that benefit.