We re a party of 4- looking for low key, authentic, fun and possibly a view?
For Monterossa al Mare:
View - Hotel Porta Roca, you will pay for the view. Low key as well, but upscale.
I'm not entirely certain there is a single venue to meet all your criteria. For great food, L'Osteria. Belvedere may be the closest to meeting all your points.
I need to understand what you mean by "fun"?
In Vernazza, try Belforte for the view & food.
Monterosso: Cantina di Miky. Ask for a table outside on the beachside/waterside deck.
I second Cantina di Miky!