One avenue this entity could take is to offer itself as a non-profit third party administrator of a self-funded plan for other companies. Large companies typically put up the money for health insurance anyway combined with employee premiums and then pay a TPA company or health insurer to create the benefit plan and administer it for them. They don't actually buy insurance as a product from a health insurer for their employees.

Alternatively, they could pool funds and create an actual insurer.

The trouble with Amazon trying to become THE payer or TPA is that insurance is highly variable across the country. There are many different laws governing insurance in each state, and you need to strike business agreements with literally countless small practices, hospital systems, lab service providers, drug companies, outpatient clinics, etc. It would actually be a massive administrative feat for Amazon to provide solid coverage across the US.