Hi Susan,
On the Adagio Payables side of things, you can create an 'Additional Form' spec that you can print/email/fax/ePrint that lists just the remittance 'advice' portion, of which invoices were paid. This can be sent to the vendor by email, and Adagio email addresses can be configured with multiple addresses separated by semi-colons ;
After a check run, whether it is submitted with TelPay, EFT directly to the bank, or even printed checks to paper the old-fashioned way, choose Yes to printing additional forms, then print/email/fax/ePrint these forms. You have to set the bank to allow additional forms.
Here's a link to a short video about
Adagio EFT Processing on our website, which includes a brief discussion at the 1:35 mark.
I don't know about the configuration in TelPay itself, in regards to email address setup.