Need pain relief for spinal stenosis. What is safe to take throughout pregnancy?

Talk to your ob. Check with your obstetrician before any treatment. Often she or he will let you take some medications (acetaminophen or tylenol is allowed by some). While walking may be uncomfortable, exercise while flexed (bent forward), as on a bicycle or recumbent bike may be comfortable, and exercise often reduces pain. In more severe cases, your OB may allow an epidural injection.
Opiates. Of course, all medications carry some risk and it's always best to avoid them. However, this is not very practical. Most of your opiates and synthetics are fda category b/c and are relatively safe.
Several options. As you get bigger with the baby altering your posture into more lumbar lordosis decreasing the canal space, a pregnancy harness may help restore more normal contour. Sometimes, an epidural steroid injection under a local anesthetic via a caudal or non guided translaminar approach may be permitted by your OB doc with some physical therapy and possibly medication like tylenol (acetaminophen).