Coordinates for these & for their four-dimensional analogs were published by HSM Coxeter, first in 1948 in Regular Polytopes, pg 52-53 (Methuen, London) and again in subsequent revisions; any/all are highly recommended reading. The table for (quasi) regular 3D polyhedra is transcribed below.
The orientations minimize the number of distinct coordinates, thereby revealing both symmetry groups and embedding (eg, tetrahedron in cube in dodecahedron). Consequently, the latter is depicted resting on an edge (Z taken as up/down).
SOLID VERTEX COORDINATES
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Tetrahedron ( 1, 1, 1), ( 1, -1, -1), (-1, 1, -1), (-1, -1, 1)
Cube (±1, ±1, ±1)
Octahedron (±1, 0, 0), ( 0, ±1, 0), ( 0, 0, ±1)
Cubeoctahedron ( 0, ±1, ±1), (±1, 0, ±1), (±1, ±1, 0)
Icosahedron ( 0, ±p, ±1), (±1, 0, ±p), (±p, ±1, 0)
Dodecahedron ( 0, ±i, ±p), (±p, 0, ±i), (±i, ±p, 0), (±1, ±1, ±1)
Icosidodecahedron (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2), ...
(±p, ±i, ±1), (±1, ±p, ±i), (±i, ±1, ±p)
p = (sqrt(5)+1)/2 (golden mean)
i = (sqrt(5)-1)/2 = 1/p = p-1