I wasn’t sure if they could or not.
I think the long-standing rule has been that a team can not block a coach from an up-grade in level.
So, a Coordinator can
not be blocked from a head coaching opportunity (Staley last year).
A position coach can
not be blocked from being a Coordinator.
However, a Coordinator can be blocked from being hired by another team as a Coordinator because that's a Lateral-Move. I believe Staley was interested in bringing O'Connell with him to the Chargers but the Rams did not want to lose both their coordinators and blocked the move.
Interesting, after the 2017 season, McVay could have blocked his OC (Matt Lafleur) from going to Tennessee as Vrabel's OC. Technically, that was a Lateral-Move but McVay realized it was a good opportunity for Lafleur to call plays and be the top-offensive-guy to a young defensive-Head Coach.
I thought that spoke really well of McVay, allowing a young coach to take advantage of a good opportunity. One year later, Lafleur had the Packers' job and has won 40 regular season games in three years.