"Online account opening" is not a "product", but rather part of the delivery systems for many types of products, deposit and credit. As such, there's no single law, rule, regulation, policy, or procedure that allows "one stop shopping." Also, remember that "opening an account" and "opening a new relationship" are not the same thing -- especially from the risk control, security, and BSA/AML perspectives. Most (all?) of the consumer protection laws/regs will apply equally, but there can be variations in the way they must be implemented. Once again, there are no shortcuts. Disclosure timing, format, and content requirements will vary from one product to the next.
Two (probably more) legal/regulatory requirements are more universal -- COPPA and ESIGN. We've discussed both of these laws extensively over the past 21 years, so there are hundreds of threads exploring them.