What are the legal requirements to manage online payments of third parties through my company's account?
I'm planning a project where e-commerce can sell and payments via Paypal/Stripe are managed by my company's accounts; which later will transfer the money to their bank account. Beyond not violating the gateway's terms and conditions (which we wouldn't be), what would be legal requirements required to do this? i.e. special financial licenses, I need to know about.
You effectively will be sitting in the same transactional chain / sphere as the likes of Paypal - so you would be bounded by similar laws, operating standards and security requirements as they do in the territory you wish operate in. If you will be holding peoples money, you become a bank with the obvious risks and vulnerabilities, so it is best you seek professional advice.
If you are receiving the funds, you are acting as the merchant; therefore your legal entity must be present on all websites. This also means you are responsible for all disputes linked to your "partners".
If your target is instead to act as a financial institution which holds the money (but the merchants need their own account), you need a banking license.
An hybrid model is not possible.
1st - many U.S. states have laws against this type of non-bank money transactions.
2nd - you could be creating liability for yourself for 'laundering" money.
I suggest that you should discuss your business model with an attorney before going into this business.