A lot of the time, if you communicate with your supplier, they will not require your SSN. If you do not want to provide your SSN, explain you are not comfortable giving out that information on an opening application, but you are more than willing to provide any business documentation required. If the supplier still asks for it, you should ask why they need it when they have everything else or move on to another supplier.
Why do I need to give my social security number to a supplier?
Do I give supplier my social security number?