Can people on H4 EAD do private tutoring?
i.e. Should the private tutoring be as part of any registered company or can we do it individually and take money in cash and pay the tax appropriately for the year?
If you have an EAD you can do work in any way you want -- e.g. you can work individually, for a company, for your own self-employed company, etc. It doesn't matter.
Of course you will need to follow the appropriate tax rules for whatever form of work you do.