How do I access the App Store on my 3rd Generation Apple TV

There is no app store on Apple TV 3rd gen.

You only get the apps already installed.

See:

Download apps on your Apple TV - Apple Support

If you can't find the App Store

The App Store isn’t available on Apple TV (3rd generation). To get new apps that you can use on your Apple TV (3rd generation), update your software.

Which devices work with HBO Max? - HBO Max Help Center

• Apple TV 4K or Apple TV HD with the latest tvOS software (i.e. 4th generation or later)

What devices and platforms are supported by Disney+? - Disney+ Help Center

• Apple TV (4th generation and later)