A more general question is: Can a witch or wizard die a natural death as long as her/his horcruxes exist?
Looking for canonical answer only. Rowling's interviews etc are welcome, too.
No, they cannot, nor can Voldemort:
‘Well, you split your soul, you see,’ said Slughorn, ‘and hide part of it in an object outside the body. Then, even if one’s body is attacked or destroyed, one cannot die, for part of the soul remains earthbound and undamaged.’
Harry Potter and the Half-Blood Prince - pages 464-465 - Bloomsbury - chapter 23, Horcruxes
The purpose of a Horcrux is to tie the soul to this world. Even if the body dies, the soul would not move on to the next plane of existence, as Harry found out when Voldemort killed him. If the body is intact, then the body is restored to its natural state. If the body is destroyed, the soul is left to wonder as a ghoul or specter, as Voldemort was until Quirrell found him.
Since it was not explained, it leaves two possibilities to what happens to aged or sick Horcrux users. Either the body gives way and they are left as a malignant spirit, or the body continues to weaken until the spirit is bound into a feeble state, essentially a corpse. In any case, as we saw in the Goblet of Fire, a new body can be created using magic to which the original (divided) spirit can then be bound to. For all intents and purposes, a Hocrux user is as immortal as the Horcrux he or she creates. Strong and able bodied is a different matter, as immortality here means bound to the earth.