Is someone who shaves daily all year round allowed to shave during the week of Succot and Pesach (on Chol Hamo'ed)?
No. Shaving is forbidden on Chol Hamo'ed. Shaving one's beard with a razor is always forbidden, but during the week of Succot and Pesach, it's even forbidden to trim it with scissors.
In the times of the Chatam Sofer (Rabbi Moshe Sofer), some Jews in the high society used to shave with a razor, disregarding Jewish Law. Now, the Torah prohibition against shaving with a razor doesn't apply until the beard stubble has grown somewhat (approximately two or three days' growth). So the Chatam Sofer said that it was better for them to continue shaving on Chol Hamo'ed so that the hair would not have time to grow to that length, rather than letting it grow during and shaving after the festival, which would entail the violation of hundreds of Torah prohibitions, one prohibition for every two hairs shaved. Many people misunderstood this, mistakenly thinking that the Chatam Sofer was lenient regarding the prohibition of shaving on Chol Hamo'ed.