Depending on the exact methods by which you want to notify the team members it should absolutely be possible; this is basic functionality of the Process Builder (or even Workflow Rules) that you can learn more about on Trailhead:
Process Automation on Trailhead:
The process builder allows you to build simple-to-moderately-complicated triggers (with some limitations). For your purpose the Process could be fired when the "Application" object is created/is updated to meet the criteria "Field is equal to 'Approved'", and specify the action to take once that criteria is met.
The action you specify can be simple, such as sending an email (you will need to create an email alert and accompanying email template), it can update other records (which can then go on to trigger their own automation actions, or recursively trigger further automation on the same record), or it can run a custom Apex class.
If you are going the relatively simpler route, an email alert, you will specify the recipients within the customization of the email alert itself - the alerts are not specific to the process and can be re-used by other pieces of automation.