What are some examples of qualified providers that will accept this type of data?
The "attestation of PCI compliance from a qualified provider" line refers to companies like these: https://www.pcisecuritystandards.org/approved_companies_prov...
Anyone with a merchant account to accept credit cards online already has a relationship with at least a QSA, and is already getting a quarterly attestation of compliance after completing a questionnaire and security scan of their server environment; it's required by PCIDSS and merchant account providers enforce it, typically providing an account with a QSA for a non-optional annual fee and imposing another fee if you fail to remain compliant.
It's just a formality having them forward that attestation to Braintree before they hand you the data dump.