No, they don't. The lipid nanoparticles they use seem to have an adjuvant effect, but they are also directly functional, and there's not anything else in the mRNA vaccines.
It's such a complicated semantic distinction, though. The formulation chemistries chosen for mRNA drug development efforts have a much smaller adjuvant effect (to allow people to better tolerate repeated administration of the agents). But here, using an (older, simpler) formulation chemistry that pisses off the immune system is useful. So is it an adjuvant or not?
So, it's otherwise functional but also chosen to have an adjuvant effect.