Emailing check or bill payment stubs is currently unavailable in QuickBooks Desktop. However, I don't want to leave you empty handed here.
You can always print PDF copies of the payment stubs then send it as an attachment when sending emails outside QuickBooks. Let me show you how to do that:
Go the the File menu and select PrintForms.
Choose BillPaymentStubs or Checks, whichever is sent through ACH.
From the list, pick the stubs you want to print before clicking OK.
On the next page, select PDF from the Printer Name drop-down the hit Print.
Pick a location or drive to where you want to save the PDF.
If you can't see the stubs on the list, re-open the transaction and select PrintLater.
Once you're done, attach the file to the vendor's email. Also, I know how important it is to have this feature handy in the program, so I'm sending a request to our developers letting them know you want this.