iOS update

Maybe you own an iPad Air (1st generation) from late 2013, without the Touch ID. This model is too old and has insufficient hardware for iOS 13 / iPadOS 13.

If you already have 12.4.5 installed, you have the 1st gen.

Please provide us with your model number to see what generation your iPad Air is. Settings --> General --> About --> Model Number.

Like I suspected you own a 2013, 1st gen iPad Air.

iOS 12.4.x is a far as that iPad model can be upgraded/updated to.

The internal hardware is not powerful enough to be able to run the latest iPadOS 13.

Sorry.

That would be an original (first generation) iPad Air. The final supported iOS version is currently iOS 12.4.5.

The original Air cannot be updated to iPadOS 13.x.

This may be your answer to the problem: https://support.apple.com/en-us/HT201435

iPad Air model?

A 2013, 1st generation iPad Air cannot be upgraded/updated beyond iOS 12.4.x.