1099 MISC

Yes, box 6 on the 1099-Misc is the correct place to report payments to non-employees for medical services.  Further, as the controller of the account/payments made on behalf of another, the law firm would be the correct entity to issue the informational return (1099-Misc).   According to the Form 1099-Misc instructions, the firm would issue a 1099-Misc for total payments made, in the course of its trade or business, to each physician or other supplier or provider of medical or health care services that were for $600 or more. 

Here is a link to the 1099-Misc instructions.  (See Box 6 description and Paragraph "Payments made on behalf of another person").