Cinque Terre is small and very popular - few hotels, and they don't need to make too much of an effort to be booked out at high prices. Have a look at this recent trip report:
https://www.flyertalk.com/forum/35183465-post44.html

You could stay in La Spezia and take the train in. At least the NH Hotel (short trolleybus ride from the station, or 20 mins or so walk) should be bookable on Chase rewards, and while La Spezia isn't the most picturesque of places, it's also not the worst place to stay.