I have been working on a web product for over a year now and I've finally gotten to the point where I've finished integrating the payment / subscription / credit card processor into the product, except now I step forward with some hesitation.
What do I need to do before I'm allowed to charge my users for the service? Are there requirements?
I am located in Southern California, USA.
There are not a lot of requirements, but here are some things to think about.
Sounds like you are building an online product/service, so shipping/returns isn't applicable. If you have finished integrating your checkout, as you say, you already have what it takes to charge your users and whoever is processing your credit card payments for you would have terms for you to follow. I suspect you are overthinking it - if people are willing to pay for your service, you are allowed to accept payments as either individual or business.
AS Kekito said, you will need those 3 things.
You also should think about "return policies," i.e. you get up to 7 days to receive a full refund. You need to keep in mind the consumer protection laws of California, I don't know what they are off hand, but I know they're one of the more consumer friendly states.
You also should make sure that you're keeping enough for taxes to California, and anywhere else you happen to have a physical presence. If you're servers are located in a Data Center in, say, Texas, research their tax laws (even contact their Secretary of State) to see if you need to file as a company there and pay taxes. Some states treat having servers in a DC in their state as having a physical presence and will require you to collect Sales Tax from people in their.