Buckthorn, specifically Buckthorn root, appears only as a woody branched root sticking out of the sand on Healer's Island. It looks more like half-buried driftwood than a live plant.
In actuality, there are many real life plant species that use "Buckthorn" in their common names. Buckthorn most often refers to plants of the Genus Rhamnus in the Family Rhamnaceae, other members of the aforementioned Family, or possibly plants from the Genus Hippophae commonly known as Sea Buckthorns. Common Sea Buckthorn (Hippophae rhamnoides), Alder Buckthorn (Frangula alnus), and Common Buckthorn (Rhamnus cathartica) all occur in areas of Scandinavia and may have been encountered by historical Vikings.
In the How to Train Your Dragon Franchise, buckthorn root is a specific ingredient in the medicinal cure for Eel Pox, along with Dagga Plant, Rock Blossom, Goat Weed, and Bloodvein Eel. It needs to be lightly roasted to release it full effects.
In reality, many of the Buckthorn species currently or historically were used in herbal medicine. Both the Alder Buckthorn and Common Buckthorn were used historically by some European cultures for its laxative effect and deworming. However, both species are also moderately toxic to humans, and incorrect dosing can cause severe gastrointestinal purgative effects. In fact, the Common Buckthorn is also called Purging Buckthorn. Alder buckthorn has also been prized for use in gun powder once carbonized into charcoal. Unripe berries of this species as well as its bark can be used to make green or yellow dyes, respectively.
Common Sea Buckthorns, of the other hand, have very edible if not very tasty orange-yellow berries. Raw berries can be eaten, but they are sharp, sour, and oily, so they often need some sort of processing to be palatable. The oil is very valuable and is used in herbal medicine and cosmetics. Sea Buckthorns prefer sandy soil and full sun to grow, and are tolerant of salty conditions. They often grow in coastal areas close to the sea. As such, the root Snotlout retrieves may have been intended by the writers to be a Sea Buckthorn.
In the episode "The Eel Effect", Snotlout is tasked with harvesting Buckthorn Root for the Eel Pox cure. He finds a branch on the beach of healer's island and pulls it out with some difficulty. Hookfang shoots a fire blast to mildly roast it before he takes it away.