Biking in Cinque Terre?

It has been a while since I did the full hike but bikes would not have worked when I did -- they are trails not paved paths -- some of them are steep and narrow and a bike would be destructive as well as inappropriate around people. The first leg from Riomaggiori to Manarola was as I recall a wide sort of paved path but I know it has been damaged in recent years and I don't know its current state of repair. I would be surprised if bikes were allowed anyway.

Yeah, bikes would not work along the coastal trails, even the flattest most developed trail that re-opened, probably does not allow bikes. Given the rough terrain on the trails, you likely would be carrying your bike most of the time.

There is a road behind the towns that you can bike, it is hilly, so not an easy ride, but could be done. The area surrounding the villages is a National Park and is supposed to have some mountain biking trails.

The popular Cinque Terre hiking trails are not a good choice for bikes, but there are biking opportunities in the area. Because the CT slopes are so steep, consider mountain biking which is very popular in Italy, often mentioned just by the acronym MTB.
This article mentions the locations of rental centers. https://www.cinqueterre.eu.com/en/cinque-terre-bike
Alltrails has an overview of trails: https://www.alltrails.com/parks/italy/liguria--3/cinque-terre-national-park/mountain-biking

In nearby Levanto there's a unique and popular biking experience through a disused railroad tunnel linking Framura, Bonassola and Levanto. Described here: https://cinqueterre-travel.com/activities/outdoor/biking/

The first leg from Riomaggiori to Manarola was as I recall a wide sort
of paved path but I know it has been damaged in recent years and I
don't know its current state of repair.

That's the stretch of the Sentiero Azzurro called the Via dell'Amore. It has recently reopened but requires a timed-entry ticket, and I can't image a bike would be allowed on it.

https://www.parconazionale5terre.it/Edettaglio.php?id=79736

No other section of the Sentiero Azzurro would be bike-able; too steep, too heavily foot traveled and/or too many stairs. (EDITING TO ADD: per previous posts, you visited the CT this past summer so should already be aware of that?) Can't speak for other trails in the park but given the terrain I can't imagine any of them would be suitable for two wheels.

I've no idea how current the info is but these two links provide info on biking Levanto>Bonassola>Framura, renting and returning the bike in Levanto.

https://cinqueterre-travel.com/activities/outdoor/biking/
https://www.cinqueterre.eu.com/en/bike-path-levanto-bonassola

And from our own Roberto some years ago...
https://community.ricksteves.com/travel-forum/italy/biking-not-hiking-the-cinque-terre

(Oops, Mike, I was still typing as you were posting; sorry for the one duplicate link!)