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, 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.