As Pete L. Clark says, if you can Veronese your line bundle then the answer is "all of them".

So a more interesting question may be: for which varieties M does *every* ample line bundle give an embedding defined by quadrics?

The best sufficient condition I know is that MxM have a Frobenius splitting w.r.t. which the diagonal is compatibly split. See Brion and Kumar's book about Frobenius splitting, and [Sam Payne's article][1], which discusses the toric case. The first shows that it's true for flag manifolds, and the second that it's not true for all Schubert varieties.


  [1]: http://front.math.ucdavis.edu/0802.4302