Among Tolkien's mythology, Fingolfin's duel with Morgoth is well known. Morgoth suffered serious injury during the fight, and was left with permanent scars. He had also previously suffered burns from the Silmarils when he stole them.
Is there any account of another Vala being phyiscally injured, or even an indication that they could be? If not, why is this unique to Morgoth and why would he take a physical form that would make him vulnerable to mortals?
Morgoth is vulnerable to physical harm because he is only a small part of the Vala Melkor once was, locked to his original physical form. Most of his power passed in Arda itself, to dominate its matter and its habitants, to the point that the World itself became Morgoth's Ring:
Sauron was 'greater', effectively, in the Second Age than Morgoth at the end of the First. Why? Because, though he was far smaller by natural stature, he had not yet fallen so low. Eventually he also squandered his power (of being) in the endeavour to gain control of others. But he was not obliged to expend so much of himself. To gain domination over Arda, Morgoth had let most of his being pass into the physical constituents of the Earth – hence all things that were born on Earth and lived on and by it, beasts or plants or incarnate spirits, were liable to be 'stained'. Morgoth at the time of the War of the Jewels had become permanently 'incarnate'.
Other Valar (plural of Vala) did not expend their power so freely and only use bodies as some kind of clothes, as described in The Silmarillion:
Moreover their shape comes of their knowledge of the visible World, rather than of the World itself; and they need it not, save only as we use raiment, and yet we may be naked and suffer no loss of our being.
Since we are not even sure than these "clothes" are tangible and any Vala can switch back to his Spirit form at will, that makes them far less vulnerable to physical harm than Morgoth is.
The Valar took on physical forms after entering Arda, so I suppose they could be injured as any other physical being. The difference is that they could change form at will, while Morgoth, after destroying the Trees, lost that ability, so he had to suffer for his wounds permanently. In the Book of Lost Tales there's one instance in which Melkor plans to kick Tulkas in the mouth, so it's supposed that he could suffer. And Melkor gets a punch and a broken lip long before he turns into Morgoth and starts losing his powers. However, Tolkien changed many things from the Book of Lost Tales in the later Silmarillion, so I'm not sure if he kept these ideas or not.
After the creation of Arda, many of the Ainur descended into it to guide and order its growth; of these there were fifteen more powerful than the rest. Fourteen of these great Ainur became the Valar, or Powers of Arda.