I can't say that I've been following Danish news, but I can tell you for sure that for example in Italy hospitals and other institutions have asked for Covid vaccination in order to stay employed.
I wouldn't be surprised to see that Denmark does the same, and more, if vaccinations number don't reach the thresholds needed to prevent healthcare systems from collapsing.
And you should know that Denmark is far from as "laissez faire" as many Americans imagine. For example even before Covid they were enforcing some very harsh social measures targeting self-isolating communities, in order to maintain social cohesion, for example.