Yes, sir, a well-established reef tank such as your 106-gallon reef system is the type of environment the delicate dragonface pipefish need in order to do well.
As you know, pipefish in general are a bit more delicate than seahorses, and the dragon pipes are no exception. This is mainly because the pipefish are collected from the wild — unlike domesticated seahorses, they haven’t been captive bred and raised for generation after generation and they are therefore not adapted for aquarium life or accustomed to eating nonliving foods.
For this reason, the dragon pipefish are usually recommended for experts only who can provide them with an aquarium of the type that is optimized to meet the specialized needs and requirements of seahorses and other syngnathids. You have just the right type of setup for these pipefish, so I don’t see any reason for you not to give them a try. They normally ship well enough and acclimate without undue difficulty, but feeding them can sometimes be a problem.
They have a very small mouth and are adapted for feeding on micro-crustaceans, such as copepods and the smallest amphipods. Their diet does include the dreaded "red bugs" and they have a reputation for eating them, so dragon pipefish may indeed be helpful in controlling the numbers of these pests in a reef system. Dragonface pipefish are reef safe and should get along with other small reef-safe fish.
Dragonface pipefish have very long, slender, snakelike bodies and they will wrap their tails around corals and cling to them while they are feeding. Corals with powerful stings and anemones can therefore be harmful to the dragon pipes, just as they would be to seahorses, but if your reef tank only houses SPS corals, then that should not present a problem.
However, be advised that alligator pipefish are an entirely different species from the dragonface pipefish, so don’t get the two confused.
Also, Grant, if you’re having trouble with the red bugs in your reef tank, be sure to check in with the folks at Reef Central — the reef keepers on their forums can provide you with a lot of additional information on eradicating the red bugs that you should find very helpful.
Best of luck with your reef tank and the pipefish, sir!