*You* don't use a 1099 form of any type. If no W-2 was issued to you, then as far as the IRS is concerned you earned that money while self-employed and in business for yourself. So all of your business income and business expenses will be reported on SCH C as a part of your personal tax return.
Ask the business owner if they plan to issue you a form 1099-Misc.
Whether they do or not, you will, as Carl said, report the income as self employment on schedule C or C-EZ.
Although it may be news to you that you “have a business” (and most of the other people who get their first 1099-MISC); The IRS considers anything in box 7 of 1099-MISC to be self employment income. Enter the 1099-Misc income, at Other Common income / income from a 1099-MISC . TurboTax (TT) will complete Schedule C or C-EZ for you and allow you to deduct any expenses associated with this income. You'll also have to pay self employment tax (social security & Medicare) on any profits greater than $400. These types of payments are frequently done this way. Your "employer" is not the only one.
The IRS also considers undocumented cash income (no W-2 or 1099-MISC), for work performed, to be self employment income.