Hi there, @brian233.
The business name will automatically appear in the Payer's Name field on the 1099 form. This is your registered business name as specified in the Company Information section under Account and Settings.
While the owner's/contractor's name should be reflected in the Recipient's Name field, not on line one, since this section reports the primary gross income or distribution before taxes or deductions of that contractor.
You can refer to the screenshot below:

You can also read this article to further help you in preparing this tax form: Create and file 1099s with QuickBooks Online.
It is also worth noting that if you're not the middle contractor of the sole proprietor and you don't pay them $600 or more from your own business account, there's no need for you to prepare the 1099 for them. It is the employer's obligation to file it.
If you have additional questions, let us know by leaving a reply below. We'll assist you promptly.