Yes, there are many handicapped Elves, though most of them are not narratively important.
You mention Maedhros, who lost his hand and had to do without it. There is no indication that the Elves could have done anything to heal his wound, and there was nothing 'magical' about it; Fingon is the one who cut his hand off. Elves do seem to heal more quickly than Men, and to generally be more resilient, but only to a limit. Indeed Maedhros's recovery abilities from his torment (aside from his amputation) appears to be exceptional, something that only the Elves who were 'nurtured in Valinor' might have.
There Maedhros in time was healed; for the fire of life was hot within him, and his strength was of the ancient world, such as those possessed who were nurtured in Valinor. His body recovered from his torment and became hale, but the shadow of his pain was in his heart; and he lived to wield his sword with left hand more deadly than his right had been.
(Silmarillion, Of the Return of the Noldor)
Gwindor is another example:
At first his own people did not know Gwindor, who went out young and strong, and returned now seeming as one of the aged among mortal Men, because of his torments and his labours; and now also he was maimed.
(Children of Hurin; Turin in Nargothrond)
We are not told of the exact nature of Gwindor's handicap, so far as I know. But it's hard to believe Gwindor was the only Elf that Morgoth tormented in that manner.
In fact, Gwindor's brother, Gelmir, had been blinded by Morgoth (and was the reason for Gwindor's capture):
With them they brought Gelmir son of Guilin, a lord of Nargothrond, whom they had captured in the Bragollach, and had blinded; and their heralds showed him forth crying: ‘We have many more such at home, but you must make haste if you would find them. For we shall deal with them all when we return, even so.’ And they hewed off Gelmir’s arms and legs, and left him.
(Children of Hurin, Battle of Unnumbered Tears)
I think it's safe to say that whatever torture and maiming you can conceive of was done to the Elves by their enemies. It's not just something that can hypothetically happen to them.