I have e-filed my first 1099-NEC form to a single-member LLC. I put LLC's name under recipient's name instead of the owner's full legal name. Is that a mistake?

If you paid the LLC (wrote the check to the LLC), then you entered the name correctly. 

What if i paid via Zelle money transfer?

The same rule applies, if the Zelle was sent to his business account under the LLC then you fill out the 1099-NEC to the business name. 

Either way you entered it, you can still take the deduction as contract labor expense. It will matter more on their end how it was filed. 

Going forward, make sure to have every contractor fill out a W9 Form before they begin work. This way they can enter the correct name and Tax ID number they prefer. And you will have all the info necessary to file the 1099-NEC after the year ends. 

Thank you for your message.  I did get a W9 from with my contractor's full name as well at an LLC's business name they operate under. In my 1099-NEC form I only wrote their LLC business name without their first and last name. Would that be a problem?

It should be fine to use the LLC name on the Form 1099-NEC as it will match the EIN they put on the W-9 form for you.