`Polyhedron`s are a very new data type and it is not unusual that not all possible functions have been overloaded for them.

You can extract the vertex coordinates of a `Polyhedron` `R` with `R[[1]]`; combined with `ConvexHullMesh`, this allows you to convert the `Polyhedron` to a `MeshRegion` and to apply `RandomPoint:

    R = RandomPolyhedron[5];
    S = ConvexHullMesh[R[[1]]];
    RandomPoint[S]

With the user-defined "polyhedron", you can do, e.g., this:

    R = BoundaryDiscretizeGraphics@Graphics3D[
       Polygon[{{{0.9000000000, 
           9.803074361, -0.2788974201}, {0.6750000000, 
           10.19278579, -0.2270388454}, {0.4500000000, 
           9.803074361, -0.2935999100}}, {{0.6750000000, 
           10.19278579, -0.2270388454}, {0.2250000000, 
           10.19278579, -0.2379129589}, {0.4500000000, 
           9.803074361, -0.2935999100}}, {{0.2250000000, 
           10.19278579, -0.2379129589}, {0, 
           9.803074361, -0.3066046725}, {0.4500000000, 
           9.803074361, -0.2935999100}}, {{0, 
           9.803074361, -0.3066046725}, {0.2250000000, 
           9.413362929, -0.3638887807}, {0.4500000000, 
           9.803074361, -0.2935999100}}, {{0.2250000000, 
           9.413362929, -0.3638887807}, {0.6750000000, 
           9.413362929, -0.3472567826}, {0.4500000000, 
           9.803074361, -0.2935999100}}, {{0.6750000000, 
           9.413362929, -0.3472567826}, {0.9000000000, 
           9.803074361, -0.2788974201}, {0.4500000000, 
           9.803074361, -0.2935999100}}, {{0.9000000000, 
           9.803074361, -1.278897420}, {0.6750000000, 
           10.19278579, -1.227038845}, {0.4500000000, 
           9.803074361, -1.293599910}}, {{0.6750000000, 
           10.19278579, -1.227038845}, {0.2250000000, 
           10.19278579, -1.237912959}, {0.4500000000, 
           9.803074361, -1.293599910}}, {{0.2250000000, 
           10.19278579, -1.237912959}, {0, 
           9.803074361, -1.306604673}, {0.4500000000, 
           9.803074361, -1.293599910}}, {{0, 
           9.803074361, -1.306604673}, {0.2250000000, 
           9.413362929, -1.363888781}, {0.4500000000, 
           9.803074361, -1.293599910}}, {{0.2250000000, 
           9.413362929, -1.363888781}, {0.6750000000, 
           9.413362929, -1.347256783}, {0.4500000000, 
           9.803074361, -1.293599910}}, {{0.6750000000, 
           9.413362929, -1.347256783}, {0.9000000000, 
           9.803074361, -1.278897420}, {0.4500000000, 
           9.803074361, -1.293599910}}, {{0.9000000000, 
           9.803074361, -0.2788974201}, {0.9000000000, 
           9.803074361, -1.278897420}, {0.7875000000, 
           9.997930077, -0.7529681327}}, {{0.9000000000, 
           9.803074361, -1.278897420}, {0.6750000000, 
           10.19278579, -1.227038845}, {0.7875000000, 
           9.997930077, -0.7529681327}}, {{0.6750000000, 
           10.19278579, -1.227038845}, {0.6750000000, 
           10.19278579, -0.2270388454}, {0.7875000000, 
           9.997930077, -0.7529681327}}, {{0.6750000000, 
           10.19278579, -0.2270388454}, {0.9000000000, 
           9.803074361, -0.2788974201}, {0.7875000000, 
           9.997930077, -0.7529681327}}, {{0.6750000000, 
           10.19278579, -0.2270388454}, {0.6750000000, 
           10.19278579, -1.227038845}, {0.4500000000, 
           10.19278579, -0.7324759022}}, {{0.6750000000, 
           10.19278579, -1.227038845}, {0.2250000000, 
           10.19278579, -1.237912959}, {0.4500000000, 
           10.19278579, -0.7324759022}}, {{0.2250000000, 
           10.19278579, -1.237912959}, {0.2250000000, 
           10.19278579, -0.2379129589}, {0.4500000000, 
           10.19278579, -0.7324759022}}, {{0.2250000000, 
           10.19278579, -0.2379129589}, {0.6750000000, 
           10.19278579, -0.2270388454}, {0.4500000000, 
           10.19278579, -0.7324759022}}, {{0.2250000000, 
           10.19278579, -0.2379129589}, {0.2250000000, 
           10.19278579, -1.237912959}, {0.1125000000, 
           9.997930077, -0.7722588157}}, {{0.2250000000, 
           10.19278579, -1.237912959}, {0, 
           9.803074361, -1.306604673}, {0.1125000000, 
           9.997930077, -0.7722588157}}, {{0, 
           9.803074361, -1.306604673}, {0, 
           9.803074361, -0.3066046725}, {0.1125000000, 
           9.997930077, -0.7722588157}}, {{0, 
           9.803074361, -0.3066046725}, {0.2250000000, 
           10.19278579, -0.2379129589}, {0.1125000000, 
           9.997930077, -0.7722588157}}, {{0, 
           9.803074361, -0.3066046725}, {0, 
           9.803074361, -1.306604673}, {0.1125000000, 
           9.608218645, -0.8352467266}}, {{0, 
           9.803074361, -1.306604673}, {0.2250000000, 
           9.413362929, -1.363888781}, {0.1125000000, 
           9.608218645, -0.8352467266}}, {{0.2250000000, 
           9.413362929, -1.363888781}, {0.2250000000, 
           9.413362929, -0.3638887807}, {0.1125000000, 
           9.608218645, -0.8352467266}}, {{0.2250000000, 
           9.413362929, -0.3638887807}, {0, 
           9.803074361, -0.3066046725}, {0.1125000000, 
           9.608218645, -0.8352467266}}, {{0.2250000000, 
           9.413362929, -0.3638887807}, {0.2250000000, 
           9.413362929, -1.363888781}, {0.4500000000, 
           9.413362929, -0.8555727816}}, {{0.2250000000, 
           9.413362929, -1.363888781}, {0.6750000000, 
           9.413362929, -1.347256783}, {0.4500000000, 
           9.413362929, -0.8555727816}}, {{0.6750000000, 
           9.413362929, -1.347256783}, {0.6750000000, 
           9.413362929, -0.3472567826}, {0.4500000000, 
           9.413362929, -0.8555727816}}, {{0.6750000000, 
           9.413362929, -0.3472567826}, {0.2250000000, 
           9.413362929, -0.3638887807}, {0.4500000000, 
           9.413362929, -0.8555727816}}, {{0.6750000000, 
           9.413362929, -0.3472567826}, {0.6750000000, 
           9.413362929, -1.347256783}, {0.7875000000, 
           9.608218645, -0.8130771013}}, {{0.6750000000, 
           9.413362929, -1.347256783}, {0.9000000000, 
           9.803074361, -1.278897420}, {0.7875000000, 
           9.608218645, -0.8130771013}}, {{0.9000000000, 
           9.803074361, -1.278897420}, {0.9000000000, 
           9.803074361, -0.2788974201}, {0.7875000000, 
           9.608218645, -0.8130771013}}, {{0.9000000000, 
           9.803074361, -0.2788974201}, {0.6750000000, 
           9.413362929, -0.3472567826}, {0.7875000000, 
           9.608218645, -0.8130771013}}}]
       ];
    RandomPoint[R]

PS.: I have no clue at all what this new datatype is supposed to ship what is not already provided `MeshRegion`s.