How do I install vinyl shutters right where a gutter downspout is located? Do I skip the shutter there? Do I remove the gutter and relocate it? What looks best?

Id skip the shutter unless you can move the gutter

Disconnect the downspout and install you shutter. Reinstall the down spout and it should have enough flex to be out over the shutter. It may look better if you could reroute the downspout.