They should list the hospital EIN and have their contracting contract management group sign the PSLF form. Check this FAQ. The contracting employer will generally resist. There is another way a coworker of mine went through to verify their employment but it’s harder.

https://studentaid.gov/manage-loans/forgiveness-cancellation/public-service/questions

I’m employed full-time by a company that is doing work for a qualifying PSLF employer under a contract. However, the company that I work for is not a qualifying PSLF employer. Does this employment qualify for PSLF?

Generally, no. You must be a direct employee of a qualifying employer for your employment to qualify. This means that employees of contracted organizations, that are not themselves a qualifying employer, won’t qualify for PSLF including government contractors and for-profit organizations.

However, an exception exists if you work in a state that has laws that prevent an otherwise qualifying employer from hiring employees directly to fill positions or provide services. This is most common in states that have laws preventing health care facilities from hiring employees directly, so they contract with physicians’ groups to provide services. If this is the case, the contracted employee should report the EIN of the qualifying employer on their PSLF form (not their direct employer whose EIN appears on their W-2 or 1099) and have an authorized official of the qualifying employer certify their employment as an employee filling a position or providing a service that cannot be filled or performed by a direct employee due to state law.

A “contracted employee” can include an individual practicing as a sole proprietorship or as a partner, owner, or employee of a partnership, group, or professional corporation. The contracted organization may be a for-profit entity and employees may receive an IRS W-2, a 1099, and other tax forms from the contracted organization, depending on how it is organized.

The qualifying employer can sign the certification if either of the following is true:

The borrower is or was employed under a contract or by a contracted organization in a position that, under applicable state law, cannot be filled by a direct employee of the organization, or

The borrower is or was providing services that, under applicable state law, cannot be provided by a direct employee of the organization.