Can a vanilla Ubuntu 16.04 LTS Server run without snapd?

snapd is seeded in the default install so as to enable snaps to be installed without further work. However, no part of the base install is a snap (you can verify via snap list, it should return no snaps). Because of this, snapd can be safely removed with no ill side effects (Update: As of 18.04 this isn't the case, various apps such as the calculator are snaps, you'll need to install the deb version of them if you want them back):

sudo apt purge snapd

It will probably leave some dependencies lying around. You may want to remove them as well:

sudo apt autoremove

Bug #1642068 was logged against this issue and has recently been fixed: https://bugs.launchpad.net/ubuntu/+source/snapd/+bug/1642068