Yes, teleporting counts as moving, follows the normal steps for movement (except for the Walls step), and would trigger the Mangler Caltrops during the Enter Zone Effects step. RAW this is true even if you teleport the creature 0 zones (i.e. back into the zone it started at) triggering both the Leave Zone Effects and the Enter Zone Effects of that zone. These rules can be found on page 10 of the Core Rulebook.
There is a rule that being "put into play" does not count as entering the zone, which keeps creatures from suffering Enter Zone Effects when Summoned, but does not protect them when teleporting. This can be found in the Supplement on page 9.