Can I use my social security number on an invoice if I'm operating as a sole proprietorship or do I need to apply for a Tax identification number?

You can use your social security number on an invoice if you are a sole proprietor.

A sole proprietor can get an EIN number.

For more info see: Do You Need an EIN?

But in the age of identity theft---do you want to put your Social Security number out there for the world?  Get a Tax ID.