We weave our nets from the fibers
The water is sweet inside
We use the leaves to build fires
We cook up the meat inside
Now, my title might not be 100% fair to the film, as we are clearly shown workers digging gardens of some kind (probably, given what I know of Polynesia, for sweet potatoes) and even in the song that says "We use each part of the coconut / That's all we need" they refer (as I showed above) to cooking meat, probably pork (given what I know about Polynesia - and also the surprisingly dark running gags about eating poor Pua).
Still, though, the coconut palm is obviously crucial to their existence, so I was wondering if any real Polynesian culture utilized it to such a great extent. It should be noted that Moana's intention to accurately depict ancient Polynesia is about as strong as Tolkien's was to depict medieval England, so creative liberties are expected, but still.
Yes; through-out Polynesia people used the coconut and coconut tree extensively in the manufacture of just about everything possible. The song encapsulates some key uses-- coconut fiber was used to make rope that could be used for nets, fishing line, for rigging on the canoes and this rope was so tough and seemingly durable, that even colonial authorities, I think of the French in the Tuamotus, who thought the fiberous rope might be an excellent export. Yes, leaves could be used for fire, but they were also woven into mats, baskets, walls, roofs, plates, fans... just like the uses of the fiber, the list can really go on, though sometimes it depends on which coconut tree we are talking about since their leaves can be a bit different.
The coconut itself was of course very useful; the fibers inside the shell were perfect for kindling to actually get a fire started (which could be done rubbing wood from the trees-- check youtube to find demonstration videos for most of these coconut related uses). The meat of the coconut could be eaten, it could be dried, it could be squeezed to get coconut milk, etc. And of course the water inside was also very useful-- useful on long voyages as coconuts could fill a dual role as food and beverage rations.
Whether or not an island was dependent on coconut really depends on what kind of island it is. High volcanic islands such as Tahiti or really any one of the Hawaiian islands have lots of other crops/plants/sources for the material that the coconut can provide, yet, coconut groves were widely planted. This perhaps indicates a preference for the raw materials of the coconut tree, or, simply a testament to its versatility when needed. The trees also require little maintenance or care, so its a perfect if you labour is needed elsewhere as you don't need to constantly be tending it. In fact, coconut trees would not be an omnipresent across the Pacific, or exist in such densities, if it weren't for people carrying them and planting them. Establishing coconuts could be a key part of terraforming an island and actually making it possible for human settlement.
On low islands, such as the Tuamotu archipelago, a chain of atolls, coconuts were the crop that sustained human settlement. Only one of the islands in the entire archipelago has permanent freshwater-- so every other lagoon dwelling population depended on trapping rainwater and coconuts. Most of the islands get a smattering of rain daily, but in times of drought coconuts could be the only water source (aside from sailing to Makatea where there was access to freshwater caves).
Additionally; I think you make a good guess at the crop they are planting, however, it is probably taro.
Moana fits into Pacific history as a sort of fictionalized, fairytale version of something known as the Polynesian Pause, a 500 year period or so when the settlement of islands to the East stops (roughly 500 BCE- 0 CE). This took place before the sweet potato had been acquired from South America.