I am a subcontractor who is working with a former business partner and all of the pay checks are written in his name. Can the person we are working for issue me a 1099 even though no checks were written in my name?

The person for whom you are working should issue 1099 to your business partner and your business partner should issue 1099 to you for the payment your business partner made to you.

What MukatA says is correct.

Even though the checks were never issued in your name, you did in fact receive the money. That is why your former business partner would issue you a Form 1099-MISC to document the payments to you.