Yes, this achievement is still obtainable, and here's how.
With the recent release of The Taken King, Crucible/Vanguard Marks have been eliminated and with it the weekly limits. Because of this, Bungie has changed the requirements of the achievement to simply reach Level 1 with the respective faction.
In short, the new requirements to unlock this are to reach level 1 in the Vanguard. Bounties, quests, and patrol missions will get this unlocked for you soon enough.
Here's the link to the post detailing this change:
https://www.bungie.net/7_TrophiesAchievements-Valorous-and-N...