There are a couple issues off the bat.
1. You're minimizing the 2 hardest parts of doing anything medical, which is the insurance billing and state licensing requirements. The state may consider you a medical services provider even if all you do is pair people with already-certified medical professionals.
2. "That day, some delivery service will bring you the Rx." This sounds illegal or heavily licensed/regulated. It's a drug addict's dream because it's so easy to maintain multiple identities. You probably get a random doctor, so the doctor won't notice it's you with a different name. And then they even send a driver to pick up the prescription, so the pharmacy won't notice that it's you with a different name.
The pharmacy would also likely need to be in on it. I.e. in Texas, pharmacies that deliver are required to include any information that would have been given orally as a written document with the prescription. If the pharmacy doesn't know it's being delivered, they won't know to include those.
If you wanted to partner with someone, it would be CVS or Walgreens, though. They already have licensed pharmacies everywhere, and handling the pharmacy requirements are going to be the hard part. I believe they also own a lot of those tiny clinic chains that are inside grocery stores, and this would give them a way to fill in any downtime in the nurses' schedules, as well as to let them soak up excess demand in one area with slack in another.