iPad Air 7 will not update

Which iPad model do you have?

There is no iPad Air 7Gen, and an iPad 7Gen. can be updated to the latest iOS 15.6

iOS 15 and iPadOS 15 are compatible with these devices

Check this article how to update iOS with the help of your computer, if the upgrade over Wi-Fi is not working:

Update your iPhone, iPad, or iPod touch - Apple Support

As described, you very likely have a first generation iPad Air.

Not sure which iPad you have? Identify your iPad model.

The first generation iPad Air, iPad mini2 and mini3 can only be updated to iOS 12.5.5.

Apple ended major update support for these models of iPad in September 2019 - although they since continue to receive occasional “point” security updates - the most recent being released late last year. These devices cannot be updated to iPadOS 13 (or any later major versions of iPadOS) as the internal hardware does not meet the minimum technical requirements (CPU and RAM) for newer versions of iOS/iPadOS.

Some App developers have dropped support for iOS versions preceding iOS 13.x for technical reasons (often because APIs or software libraries upon which they rely have been updated in the new version of iPadOS, making their App incompatible with earlier versions). 

Whilst many App titles remain compatible with your iPad, if you have one of these three models and you need to access to Apps requiring the most recent versions of iOS, you’ll need to consider replacement of your iPad with a newer or current model that supports iPadOS15.

Whether or not you are able to update any iPad to a newer version of iOS/iPadOS will be dictated by the iPad model - and the currently installed version of iOS.

Here is a list of the highest version of iOS supported by the older models of iPad (for simplicity - “generations” are shortened to a number) that are restricted in their ability to be updated:

iPad1 - iOS 5.1.1

iPad2, iPad3 and iPad mini1 - iOS 9.3.5 (WiFi Only models), iOS 9.3.6 (WiFi & Cellular)

iPad4 - iOS 10.3.3 (WiFi Only), iOS 10.3.4 (WiFi & Cellular)

iPad Air1, iPad mini2 and iPad mini3 - iOS 12.5.5

Newer models, not appearing in this list, are all able to be updated to the current version, iPadOS 15.6

If your iPad is capable of being updated to a more recent version of iOS/iPadOS than is currently installed, these support pages outline both the available methods by which the update can be performed - and troubleshooting should you experience difficulties:

Update your iPhone, iPad or iPod touch – Apple Support

If you can't update or restore your iPhone, iPad or iPod touch – Apple Support

If your iPhone, iPad or iPod touch won’t update - Apple Support

iOS update and restore errors – Apple Support

As the model was correctly diagnosed as an iPad Air1, all of my subsequent comments apply.