You need to rotate that vector yes. However standard material nodes do not have required math exposed. So you need to make hlsl custom shader. It is not that hard. And coincidence i made similar shader to what you want about month ago. You can either search forum for NASA skybox texture + nawrot, or wait a bit for me to post that shader again.
Or you can try instead vertex shader and rotate sphere mesh trough material vertex shader.
faster is just giving you link to that post:
shader for rotating is last reply.
However it has a bug i wrongly used X,Y,Z vectors. I copied rotating calculations from system that uses Y as vertical axis (unreal uses Z), so some fiddling with calculating rotations would be required.
Also shader calculates rotation for place on surface of earth and day/year, you may want to remove those calculations. Oh NVM this version does not have that day/year calculation. But i may include updated shader per request.
ps.
for skybox that uses reflection vector you need actual CUBEMAP texture.