There are three different types of account:
The individual developer account. You can push test builds to specific devices you register with Apple. You can push to 100 or 200 (I forget which) unique devices per year. Test builds expire after three months. TestFlight is very helpful for managing the process. When you are done testing the application, you can submit it to the App Store for use by the general public, or submit it for use by a specific client through the B2B system if they want to pay per-seat costs.
The company developer account. This is like the individual developer account, except you can configure it for multiple developers.
The enterprise account. This is like the other two accounts, except you can distribute to anybody within your organisation without registering their devices, you don't have the device limit, and the builds expire once per year. You cannot submit to the App Store with this account.
So, the relevant factors are:
I am currently working on an app for a local hospital.
If it's for internal use by the hospital, then you want the enterprise account, unless you want to sell them copies of the app rather than charging them for development, in which case you'd use the B2B programme.
If it's for use by the general public, you want one of the first two accounts.
I don't know if I can use this account to push a rough draft to about 40 different devices
Yes, so long as you register their UDIDs with Apple or you use the enterprise account.
I'm afraid the app may get rejected.
You don't need to go through the approval process for test builds, but you are limited to who you can distribute them to.
You don't need to go through the approval process with the enterprise account, but you can't distribute to anybody but members of your organisation.
If the app is for internal use, you probably won't make it onto the public App Store, but it would be suitable for the B2B programme, in which case it would go through the approval process when you are ready to distribute the final version.