To the best of my knowledge this is a very hard problem and the answer to this question is, unfortunately, open.
A famous example that illustrates this in the context of quantum integrability comes from the one-dimensional Hubbard model in condensed-matter physics. Its (quite complicated) $R$-matrix was known since the late eighties, yet the corresponding quantum group was only found in the last decade, "by accident" in a very different context. Namely: in the reverse process -- the computation of the $R$-matrix for the quantum group associated to a certain Lie (super)algebra -- the result turned out to be the $R$-matrix of the Hubbard model; here the choice of the Lie algebra was motivated by string theory, and more precisely the so-called AdS/CFT correspondence. For a bit more about this, and further references, see for example (the introduction of) arXiv:1509.06205.