In every reference I see, the Seifert-Weber space is presented as an identification space (specifically identify opposite faces of a dodecahedron by a 3/10ths twist).
What I can't seem to find is a surgery diagram for this manifold. From its homology we know it doesn't arise as surgery on a knot. Its volume is approximately 11.1991. Checking with SnapPy there are a lot of candidate links in this range:
In : len(filter(lambda x: x.num_cusps() >= 3,HTLinkExteriors[11.19:13.0])) Out: 1103 In : len(filter(lambda x: x.num_cusps() >= 3,HTLinkExteriors[11.19:12.0])) Out: 365 In : len(filter(lambda x: x.num_cusps() >= 3,HTLinkExteriors[11.19:12.5])) Out: 620
so my attempts at guessing have been unsuccessful.
One could calculate a surgery diagram by-hand from the Heegaard diagram that comes from the identification space and then try to simplify the result. This seems daunting to do by hand, but I'm also unaware of software for the task.
I accepted Daniel's answer before working through the details. The process certainly gives the Seifert-Weber space a surgery on a manifold. The problem is that the branched-covers of $S^3$ over the trivial 2-component unlink are not themselves link complements, as in the case of a 1-component unlink, so the method in Rolfsen doesn't yield surgery on a link in $S^3$, but somewhere else. That somewhere else has non-separating 2-spheres in it, illustrated by this picture:
The cut disks are indicated by red and blue, the two light grey wireframe balls are the complements of the two knot components after cutting along the disks. As long as at least two of these gadgets are glued together light-to-dark, the yellow curve will pierce the solid grey 2-sphere only once.