No, when using router-advertisements the router has no way of knowing wat clients are there so it cannot updates your DNS forwarder. They just pick up the RA, create their config and start using it.
I guess if you want the DNS part to work you have to use DHCPv6, but I never tried if that works.
That is not entirely true, there might be no automatism (at least not in the exact same way as with IPv4), but every host in an IPv6 subnet is well aware of its neighbors and their addresses via NDP.
Regarding the original question: This setup is also something I want to achieve in the long run; it is of quite low priority though, thus I haven’t investigated too much yet. In short it should work via mDNS. How one would implement that with VyOS (so that mDNS announcements are taken over into static DNS) I can’t tell (yet) 
EDIT: At least 2024 it was not yet possible to resolve mDNS from VyOS, IMHO implying VyOS can’t do anything with it (besides relaying).