From what I know, the "Vanquish Streak" XP bonus is only obtainable in Multiplayer matchmaking though your actual streak (that appears in your stats room) is tracked in both Multiplayer and Solo Play in game modes such as Team Vanquish, Vanquish Confirmed, Gnome Bomb, Turf Takeover and so on. However, I once saw my sibling complete this quest on solo play so I think it may work on some "Team Setup" combinations while others not as much. Try achieving this against more offensive characters such as Soldiers or Peashooters and see if it works. I think PopCap set a setting to prevent an easy complete though I am not entirely sure.
-VoltScience: Exposition Mage