Curious About Ritual Handwashing

I am a non-Jew but am fascinated by Jewish culture, particularly Hasidic Jews since they are one of the groups that seem to follow Judaism as closely as possible to the Torah and are very tight knit. I recently learned about ritual hand washing using a two-handled cup and was wondering if Jews who follow the practice are allowed to use soap to wash their hands while using the cup or if they are permitted to wash their hands using a faucet with soap before or after the ritual washing? I’d really appreciate any insights about this since I can’t seem to find a clear answer online.

*Correction: I was informed that traditional Judaism is not necessarily synonymous with Orthodox Judaism, please excuse my initial post.