The general theory (due to Jimbo) is that each irreducible finite dimensional representation
of the quantised enveloping algebra of a Kac-Moody algebra (not of finite type) gives a trigonometric R-matrix. There is substantial information on these representations but the R-matrices are not explicit. There is a special case which is explicit and is given by the
"tensor product graph" method (this was worked out by Niall MacKay and Gustav Delius).

I used this in my paper:
R-matrices and the magic square. J. Phys. A, 36(7):1947–1959, 2003.
and you can find the references there.

If you want to go beyond this special case and be explicit then you can use
"cabling" a.k.a "fusion".

The only papers which deal with R-matrices not covered by the tensor product graph method that I know of are

Vyjayanthi Chari and Andrew Pressley. Fundamental representations
of Yangians and singularities of R-matrices. J. Reine Angew. Math.,
417:87–128, 1991.

G´abor Tak´acs. The R-matrix of the Uq(d(3)4 ) algebra and g(1)2 affine
Toda field theory. Nuclear Phys. B, 501(3):711–727, 1997.

Bruce W. Westbury. An R-matrix for D(3) 4 . J. Phys. A, 38(2):L31–L34, 2005

Deepak Parashar, Bruce W. Westbury
R-matrices for the adjoint representations of Uq(so(n))
arXiv:0906.3419

The Chari & Pressley paper deals with rational R-matrices.
The last preprint was an incomplete attempt to try and find the trigonometric analogues of these R-matrices.