The vines were never given a proper name in canon. The term "sucker vines" was used by "Yesteryear" writer D.C. Fontana, who commented, "The sucker vines were meant to be more subtle [than the danger posed to Vulcans by a creature like the Le-matya]. They lie quietly on the ground until some source of liquid wanders within their reach. Then they rise up, seize it, and drain it of its fluid. (This point was less clear in the film, due to NBC censorship but it is in the script.)" (Babel #5; Enterprise Incidents, number 11, p. 27)
In the script of "Yesteryear", Young Spock's encounter with a sucker vine is described thus; "He does not notice a dark coil of vines on the sand. As he approaches, the vines suddenly uncoil, reaching out at him, grabbing with tendrils at his legs. He dodges the clutching vines... one just missing catching him... and continues on OUT OF SHOT. The writhing, leafy vines retreat and begin to coil again, waiting for less elusive prey. (NOTE: It will be clear these are not snakes.)"