Is providing an online service just enough?
Are you referring to stock/forex/commodities/crypto trading?
These rules differ from country to country.
Can you be more specific?
This is really a question for a trade association like FINRA, not a bunch of entrepreneurs who aren't stock traders. You must be registered with the commission that regulates the exchange in which you will be dealing. You will also need a business license. And you may need to have a certification or license from another governmental regulatory body, depending on which country(ies) are involved in trading.
As far as documentation you must show or keep, that varies by country and trading floor, but it is very specific. Trading online is another hurdle beyond just trading. If you're already a Forex trader as it says in your profile, you should know many of the rules. Not only do you need documentation, but also documented mechanisms to prevent money laundering, fraud, etcetera.
What exacty will you be trading?
"Trading Company" can mean many different things. In my industry, it is someone who acts as a middle man between a factory that manufactures products and a Business that wants to buy those products to resell.
So, need to be more clear.