Cannot Access App Store with Mountain Lion

Try upgrading to El Capitan first from the following Apple link.

Your web browser is used to obtain the El Capitan upgrade.

Then, assuming that El Capitan gives you access to the App Store,

upgrade to High Sierra (or if you have a mid 2012 MBA to Mojave).

All links to El Capitan, High Sierra and Mojave are in the following link

How to download and install macOS – Apple Support (UK)

It is currently down for everyone on older OS X versions. iOS 6 devices and previous cannot acces it either. On older versions the app store is very hit and miss and your best bet is to use a USB with a later version's DMG on it.