Wizards and witches in the Harry Potter Universe should not require wands to do magic.
- Magic folk are hosts of magic from birth. Both muggles and squibs are not endowed with magical potential, only magic folk are. Somehow, magic folk are genetically different; they are created as vessels for magic channeling. A wand in the hand of a squib or muggle would be useless, so magic must not be fundamentally attached to a wooden stick.
- Harry Potter could do magic before he even knew he was a wizard. He vanished the glass in the zoo and popped up into a tree—magically. Clearly, no wand was required to do that magic.
- House elves, genetically magical creatures, can do magic without a wand. In fact many beings and creatures, thestrals: invisibility, phoenix: teleportation and uncanny strength, ext. can perform magical feats without wands.
- Many objects themselves can perform magic, the paintings in Hogwarts for instance, Hermione’s purse and the Weasley’s tent. All things which are magical in their own right and require no external force or device to trigger the magic.
It follows from this evidence that a great witch or wizard should have no trouble performing magic without a wand. I would go so far as to say that a wand becomes a crutch, a debilitation for without a wand, a wizard would assume they are powerless. When in fact, they should be able to wield even more magical might.