What Brian says.

If you do horizontal, there should be a calculation for the blower for length/diameter of pipe and vertical/horizontal travel. If the blower is under-powered, then you may end up with accumulation of stuff inside the pipe and/or cooking smells in each room the pipe travels through. If your installer can't answer that question, then the manufacturer of the hood should be able to.

Installers would rather run it out the side of the house so they don't get blamed for roof leaks. So if your installer seems hesitant about running it vertically, that might be a sign to involve a roof contractor to do the hole, shingles and flashing. Or find someone else who can do the install you want.