A RingCentral extension is the one which represents a user, in order to send SMS from different extensions as you want, you just require an access token for each extension or user.
You can do it following way:
If you are using a private app with password flow, you can bootstrap the system with static passwords for each extension. After the initial bootstrap you can use the refresh tokens to continue to receive active access tokens.
If you have end users with UI apps, your app can present an OAuth login interface to retrieve an access token for the user which afterwards can follow the use of refresh tokens . This will work with private and public apps.
reference of this answer here