|"To make my small elves coats." Illustrations to Shakespeare's A Midsummer Night's Dream, A. Rackham, common domain|
Now I'll start by saying there are some understandable reasons why people tend to go with the insubstantial view. Firstly elves and fairies are usually invisible to humans (unless they choose otherwise or the humans have some special ability or token) and this invisibility naturally lends itself to an assumption of a lack of substance. Yeats recounts a tale of a group of the Other Crowd who wanted to play a game of hurling but were unable to touch the ball until they found a human to play the game with them; this was necessary it seemed to give them solid form in our world (Yeats, 1893). Its also true that one of the better know historic texts on elves and fairies, rev. Kirk's 'The Secret Commonwealth of Elves, Fauns, and Faires' refers to the Good People as having bodies that were like a 'condensed cloud', 'congealed air', and compares them to astral forms (Kirk, 1691). This certainly paints a picture of an insubstantial being. Many types of fairies, including the Slua Si and the Wild Hunt are also described as traveling through the air, a feat which is easier to reconcile if they are assumed to be intangible.
However, that being said even Kirk goes on to discuss a variety of things that are done by the the Good People of a very physical nature. He mentions them using solid weapons, for example, and that they steal nursing mothers to wet-nurse fairy babies, these women being physically removed and sometimes returned unharmed years later. People who are taken and returned relate stories of living among the fairies which involve activities much like those found among humans including spinning, sewing, cleaning, and cooking. Similarly we see many Welsh tales of the Fair Family in which they are physically interacting with people, often by kidnapping women, children, or babies (Sikes, 1881). In one notable tale a boy who was taken by the fairies steals a golden ball from them and finds his way back out of Fairyland to make a gift of the ball to his mother (Sikes, 1881). It is quite clear in that story that the boy is acting in a physical way with the fairies - such as playing with the King's son - and that the golden ball is tangible and exists as something that humans can touch and take. Ballads like Tam Lin and Thomas Rhymer also offer examples of people interacting in very physical ways with the Good Folk, as do the myriad tales of Selkie wives. W. B. Yeats related a story of a woman whose mother had a friend among the Good People who similarly could be substantial and gave the woman one day an herb for protection; this was passed to her from the fairy woman's own hand (Yeats, 1893).
Perhaps the best answer to this question can be gained by looking at the Norse and Germanic evidence. Here we see that elves are considered both insubstantial and also able to take physical form. Grimm relates a story of an elf-maid who entered a house like smoke through a knothole, then married the son of the family, bore him four children, only to eventually leave the same way she had entered (Grimm, 1883). He also states that elves are strongly connected to butterflies because both are 'the product of repeated changes of form' (Grimm, 1883, p 462). In this way then it is not a matter of elves and fairies either being insubstantial or having form, but rather a matter of them being both, or at least able to be one or the other.
Ultimately folklore shows us stories of both elves and fairies that are shadowy and can pass through the physical substance of our world as well as stories where they are as solid and able to effect our world as we are. In some cases the choice between forms seems to be theirs, in others, such as Yeats hurling tale, there appears to be a more formal set of rules in play. In the end it would seem that it is true that elves and fairies are both insubstantial and tangible, and that we should not assume they are limited to either.
Kirk, R., (1691) The Secret Commonwealth of Elves, Fauns, and Faires
Sikes, W., (1881) British Goblins: Welsh Folklore, Fairy Mythology, Legends, and Traditions
Yeats, W., (1893) Celtic Twilight
Grimm, J., (1883) Teutonic Mythology volume 2