Need to issue a 1099-Misc to cleaning crew for a rental property. Can the 1099-Misc be issued using TurboTax Premier Edition?

Actually you might need a 1099NEC.  No, you can't use Premier.

There are several different ways to prepare W2 and 1099 forms depending on which version you have.

In the Online Self Employed version you use the Quick Employer Forms.  The forms are efiled to the IRS and you print and mail them to your contractors and employees.  Start an Online Self Employed version here,

https://turbotax.intuit.com/personal-taxes/online/

In the Mac Home & Business Desktop version you use the Quick Employer Forms.  The forms are efiled to the IRS and you print and mail them to your contractors and employees.  Buy Home & Business here,

TurboTax® Home & Business CD/Download 2021-2022 Tax Software for Personal & Small Business Taxes

In the Windows Home & Business Desktop program or the separate Business program you can prepare W2 & 1099 on your computer using the built in W2/1099 Reporter or online using the Quick Employer Forms. If you do them on your computer you have to mail the forms to the IRS yourself.  Buy Home & Business here,

TurboTax® Home & Business CD/Download 2021-2022 Tax Software for Personal & Small Business Taxes

How to prepare 1099Misc & W2 Forms.

(But I think this FAQ leaves out some parts and doesn't explain it well).

https://ttlc.intuit.com/community/forms/help/how-do-i-create-w-2-and-1099-forms-in-turbotax/00/25869

Assuming you have the Online Self Employed version,

Log into your account

Scroll down to Your Account and click on Show

Click on Create W-2s and 1099s

In your return, Anywhere on any screen you see....

Help me prepare and file W-2s and 1099s

Click on it and the side box will open, Then click on Quick Employer Forms in the side box.

Or from the Account Screen it will take you directly to Quick Employer Forms