Edit
If we set v4=={1,0,0}, and only consider v4,v5,v6,then
Clear["Global`*"];v1 = {v1x, v1y, v1z};v2 = {v2x, v2y, v2z};v3 = {v3x, v3y, v3z};v4 = {v4x, v4y, v4z};v5 = {v5x, v5y, v5z};v6 = {v6x, v6y, v6z};sol=FindInstance[{v4 == {1, 0, 0}, v4 . v4 == 1, v5 . v5 == 1, v6 . v6 == 1, v4 . v5 == -1/3, v4 . v6 == -1/3, v5 . v6 == -1/3}, Flatten[{v4, v5, v6}], Reals]Graphics3D[{Arrow[{{0, 0, 0}, v4}], Arrow[{{0, 0, 0}, v5}], Arrow[{{0, 0, 0}, v6}], Opacity[.2], Sphere[]} /. sol]
The equations seems too difficult to Solve or Reduce.Here we can find one solution.(still take long time).
v1 = {v1x, v1y, v1z};v2 = {v2x, v2y, v2z};v3 = {v3x, v3y, v3z};v4 = {v4x, v4y, v4z};v5 = {v5x, v5y, v5z};v6 = {v6x, v6y, v6z};eqns = {v1 . v1 == 1, v2 . v2 == 1, v3 . v3 == 1, v4 . v4 == 1, v5 . v5 == 1, v6 . v6 == 1, v4 . v5 == -1/3, v4 . v6 == -1/3, v5 . v6 == -1/3};sol = FindInstance[eqns, Flatten[{v1, v2, v3, v4, v5, v6}], Reals]eqns /. sol
{{v1x -> -(10/Sqrt[521]), v1y -> -(15/Sqrt[521]), v1z -> -(14/Sqrt[521]), v2x -> -(10/Sqrt[521]), v2y -> -(15/Sqrt[521]), v2z -> -(14/Sqrt[521]), v3x -> -(10/Sqrt[521]), v3y -> -(15/Sqrt[521]), v3z -> -(14/Sqrt[521]), v4x -> Sqrt[2/3], v4y -> 0, v4z -> 1/Sqrt[3], v5x -> 0, v5y -> Sqrt[2/3], v5z -> -(1/Sqrt[3]), v6x -> 0, v6y -> -Sqrt[(2/3)], v6z -> -(1/Sqrt[3])}}