I learned that one is not permitted to shave during Chol HaMoed or Sefirat HaOmer but see many observant people who freely shave during both periods. What is the basis for this?

Rabbi Moshe Feinstein [IM OC 1:163 and 2:96] wrote that there is a valid opinion to allow shaving on Chol Hamoed and the Omer days. Hence, even though this is not the common practice those who shave can rely on Rav Moshe's responsum.