In Fellowship of the Ring, "The Shadow of the Past", Gandalf tells Frodo:
Isildur Elendil’s son cut the Ring from Sauron’s hand and took it for his own. Then Sauron was vanquished and his spirit fled and was hidden for long years, until his shadow took shape again in Mirkwood.
So we know Sauron didn't have a body until he took shape again. But we also know, from The Two Towers, "The Black Gate is Closed", Gollum says:
Yes, He has only four on the Black Hand, but they are enough.
If I were an immortal being who could eventually form a new body when the old one is destroyed (which, to be sure, I'm not) I would create a body without such a flaw as a missing finger, especially one missing as a reminder of how I last lost. Maybe make something completely different, like a prehensile tentacle or something.
I have read The Silmarillion in addition to The Hobbit and The Lord of the Rings, and recognize there may be insufficient detail on how the Ainur can take physical form, but wondered if there was anything perhaps in other writings, like letters, that may shed light on this.