Appium’s Android portion is based on Uiautomator, which is designed to handle automating across different applications. You do not need to start a new driver for application B. Use the findElement methods and functions as you normally would.
You can test this out yourself. You can launch application B from application A manually, and if you run uiautomatorviewer (on your host machine), you will be able to retrieve UI information about application B.