The actual quest is picked up on either the Ziost Shadow (Empire) or the Gav Daragon (Republic). These are the support ships on the fleet you can travel to and where the operations are located. As for the group finder, it is an icon right by your map.
More specifically, the mission for Eternity Vault is available from an NPC in the Belsavis Operations Hangar on the Mission Deck of the Gav Daragon. The droid that gives you the mission for Karagga's Palace is inside the instance, right behind the shuttle you board at the beginning of the Op. (In the Hutta Operations Hangar). You have to be in a group to access this one. You can only take these missions one time.
There is no longer a weekly mission available for Story Mode Ops at the Mission Terminal