Every body is stiff first thing in the morning. It is part of the stretching that occurs along our spine as we sleep. The discs between the vertebra reabsorb overnight the water that gets squeezed out of them by being vertical all day long. That extra height to the disc causes the ligaments along the spine to be stressed, leading to that feeling of stiffness.
Since we are so tight first thing in the morning, we are also more liable to injure ourselves if we try to do too much too soon. It is not a good idea to roll right out of bed and onto a yoga mat. Give your body a chance to reduce the swelling in the discs. Wait at least 30 minutes before stressing the spine. Morning is not a good time for deep movements of the spine. Cat/Cow might be all you really need, so don't be trying for full Wheel or even deep Seal in your morning practice, unless you are naturally very mobile, or you have warmed up considerably.
Personally, I prefer to easy Sun Salutations or Lunge Salutations without the UpDogs (until I am really warmed up) first thing in the morning, then some Cats and
Spinal Flossing. If I am doing a Yin Yoga practice, I will avoid deep twists, flexion or extensions of the spine and work on the hips instead.
Cheers
Bernie