Level 1 - Requires 1500 Vanguard Reputation
Level 2 - Requires 2000 Vanguard Reputation
Level 3 - Requires 2500 Vanguard Reputation - Confirmed by HolyHa1fDead
Vanguard Reputation is earned in the following:
Strike Playlist rewards +10 / +12 / +14 / +25 upon completion depending on level chosen.
Patrol Missions rewards +10 / +25 upon completion.
Weekly Heroic Strike rewards +150 / +250 / +400 upon completion depending on level chosen.
Vanguard Bounty rewards +50 / +100 upon completion.
Quickest possible solution is to complete all daily Bounty challenges and complete Patrol Missions.
Another heads up; If you have a factions bond, mark or cloak (ie from New Monarchy) all your reputation will go to leveling for their store and NOT go towards your vanguard level.
By completing the weekly nightfall strike you gain a 1.2x xp modifier, this also gives some extra rep for completing missions on patrol.