You do own the business. You don't own Shipt, but you do own your own delivery business, which operates under contract to Shipt. You are in business for yourself, or self-employed (two different terms for the same thing). You have to report the income as business income. Tell TurboTax that you use your own name as your business name.
You can enter the 1099-MISC in TurboTax Online Deluxe, but if you have any business expenses to deduct against that income, you will have to upgrade to TurboTax Self-Employed, or use any edition of the CD/Download TurboTax software, in order to enter the expenses.