To set up a new Echo device, you must have an Amazon account (or create a new account). You can use your own account, but keep in mind that these Echo devices will have some access to your account. For example, "Alexa, where's my stuff?" will tell the person asking what you've ordered recently. On the other hand this can also make it easier for you to keep tabs on what the Echo devices are doing when things are not working as expected.
You must have a smartphone with the Alexa app to set up a new Echo device. This is just to get the new Echo to connect to your WiFi network and to log in to your Amazon account. Since you stated that these Echo devices are going to be sent to different places, setting up WiFi at your location does not mean it will connect to whatever WiFi network is at the destination location. You need to set it up to connect to the WiFi network at the location where it is going to be used. Once it is set up, the smartphone and Alexa app are no longer required. You may still need to use the app for certain things, such as managing Alexa's phone book and to set up routines. But if you never expect to do those things then you'll never have to use the phone or app again.