Hi Abbey, I would never EVER give my Social Security number to anyone other than a bank or hospital. In this case, I can't see why it is important in the process of becoming a member of Bellamora. I see no particular warning signs yet, but in the future I would definitely avoid any companies asking for information that they aren't privy to.
I would suggest contacting the company about your order for starters, and then if you feel the customer service was adequate, and you feel good about the interaction, hold put and perhaps they will straighten the situation out. If you don't feel secure, simply call the bank and ask for some extra tight security protocols in addition to trying to cancel or get your transaction reversed. Banks are all different so it depends on your options.
In any event, please keep us posted on the outcome.