Within the main canon of films and novelisations there's no indication that she can heal herself. She's certainly aging despite regularly singing so there's no major reason to imagine that the other powers would also work on her. On the other hand, her skin is utterly without blemish (despite walking around for hours without wearing shoes) which could imply that her healing function is autonomic, like Wolverine's.
This is conflicted by the game "Disney Infinity" in which Rapunzel's Healing Aura doesn't just work on those around her but also confers a healing bonus onto herself, suggesting that she can heal major injuries if the need arises.