Is it possible for teeth clenching cause really severe morning headaches?

Yes: Headaches in the morning are a common symptom of clenching or grinding at night. This is due yo the srain that you are applying to your muscles while clenching. You should see a dentist and have a niteguard made that is custom fit to your teeth. Don't go with the products you can buy at the store because they don't fit properly and you will most likely not wear it if it doesn't stay in.

Yes: If you are bruxing in your sleep then it is not uncommon to wake up with a headache. You should see your dentist for the construction of a night guard.

Yes: Usual treatment is dental evaluation and bite splint, but that may not address the cause. Several things including stress and structural problems such as cranial bone dysfunction and neck problems can also lead to clenching & grinding of teeth. A good history & phys exam, including osteopathic hands-on exam might help delineate the cause. Manipulation might help, even if bite splint is necessary.