In the 1099-SA section, it is asking to enter Payer Name. I am assuming this is each company that received payment through my HSA. I am only able to add one payee.

Payer is the HSA bank.  Payee is you.  All HSA distributions are treated as being paid to you, even if you used a debit card to pay for service directly with the provider.  

The Payer is the HSA plan administrator. The Payer information is located in the top left corner of the 1099-SA