They are infinite. Gunmar and 2 or 3 other Dawnguard NPCs will give random quests. Once you complete a quest for one of them, the game randomly creates another one for one of the NPCs. It could be the same NPC you just finished a quest for or one of the other NPCs. These random quests will have a lot of the same forms like "Go to location X and kill Y vampire" and "Go to location X, find Y item and return." The base game also has some of these quests, mostly from the factions you can join like the Companions, and also from Innkeepers (Go to location X and kill Y bandit.).
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