It sounds like you are not an employee, and the shop is not your employer. They are treating you as a self-employed independent contractor, not as an employee. So you not only can file as self-employed, you must file as self-employed, because that's what you are.
Being self-employed, you have to keep your own records of your income and expenses. Your income is all the money you receive by credit card or cash, including tips. Your expenses are the commissions you pay to the shop, and any equipment or supplies that you buy yourself.
Since you are not an employee, you will not get a W-2 at the end of the year. If you don't get a W-2, you don't enter a W-2 in TurboTax. So you do not need the shop's EIN, because they are not your employer.
Being self-employed, you will have to pay self-employment tax in addition to regular income tax. The net rate of self-employment tax varies depending on your tax bracket. The effective net rate will probably be in the range of 12% to 14%.
It looks like the shop is not withholding any income tax from the money that you earned. That's why you "have no record of taxes being reported" - because no taxes are being reported or paid. That means that, if you continue like this, you will not get any refund at the end of the year because you have not paid in any tax. Not only will you not get a refund, but you will owe a very large amount of tax at the end of the year, because you haven't paid any tax during the year. You should start making quarterly estimated tax payments to cover the income tax and self-employment tax that you will owe. If you live or work in a state that has state income tax, you have to make estimated tax payments to the state, too.
