Is it legal for public schools to require a student to run a webcam-video for remote classes in Massachusetts?
The first sub-question here is whether (public) schools can compel (parents of) students to acquire internet service, a computer, and a webcam. The MA Dept. of Education maintains that public schools must purchase at public expense textbooks and other instructional materials and supplies intended for use and re-use over a period of years, and computers fall in the category of materials intended for schools to purchase and use and re-use. There is no legislative authority to compel parents to purchase equipment or sign up for internet services. If there is a choice between in-person and online instruction, the legal requirement to attend school can be satisfied by in-person instruction.
The second sub-question is whether, if a child does have the technical ability to be connected to class via the internet, can they legally require the camera to be turned on? Every district has rules, so if there is a rule requiring parental consent in this situation, then parental consent is required. If it is legal at that level, there is still a legal risk to the school. Schools can generally do those things that are reasonably necessary for educating students as long as it doesn't infringe on fundamental constitutional rights, and the camera-on requirement is educationally reasonable. The risk to the school is violating the federal law FERPA, specifically a potential violation of the privacy requirements. Schools must protect personal information, which includes anything streaming from the camera (pictures, for example, are personally identifiable information). If you assume that they have an absolutely secure connection, then there is no risk of privacy violation. However, if you believe, even reasonably so, that a school practice creates a risk of breach of privacy, that still does not create a special right to avoid school.