My sixteen year old daughter wants her beauty spots removed is that possible without scarring?

Possible: A good plastic surgeon will be able to do it. But the risk of scarring sometimes is not in the skill of the surgeon. Some people have tendency to have larger scars with small incisions (hypertrophic scar, keliods). That is a risk you will have to take.

No, it is not: The only true way to remove "beauty marks' is to have them removed surgical with a scalpel and then sutured. Although the scarring should be minimal, there will be some very faint scarring.