My iPad 2 can't install the Apps Store application that is compatible with iOS 10

No.

And virtually all third party app and app developers have completely stopped supporting both iOS 9 AND iOS 10.

Virtually, almost all third party app developers have permanently removed all their older, over 8 and 9-year device compatible apps from Apple's iOS App Store.

The greater majority of third party apps have a minimum of iOS 12 and later now.

Put simply, Your nearly 10-year old, 2011 iPad 2 is obsolete!

If you need/want an iPad with an up to date iPadOS with access to current, updated apps, then you really need to seriously consider a new iPad model OR a much, MUCH “newer” , used iPad model capable of running the latest iPadOS versions.

There are plenty of newer, older iPad models that can run the latest, current versions of iPadOS.

Sorry & Best of Luck to You!

Sorry, not possible...

iPad2, iPad3 and iPad mini1 can only be updated to iOS 9.3.5 (WiFi Only models) or iOS 9.3.6 (WiFi & Cellular models).

Apple ended update support for these models in September 2016. Your iPad cannot be updated to iOS 10 or later major versions of iOS, as the internal hardware does not meet the minimum technical requirements for new versions of iOS/iPadOS. This limitation cannot be bypassed; these are 32-bit architecture devices, whilst all recent versions of iOS/iPadOS are designed for 64-bit hardware platforms. iPad2, iPad3 and iPad mini1 are obsolete.

You may wish to consider that compatible Apps for your iPad are increasingly difficult to find - as third-party developers are increasingly dropping support for older versions of iOS preceding iOS11/12/13. As older versions of Apps are removed from the Apple App Store, you’ll not be able to (re)download or install them - which may severely limit the usefulness and utility of your iPad if you should you ever need reset or restore your iPad. As such, if you want or need to access most current Apps, requiring more recent versions of iOS, you’ll need to consider replacement of your iPad with a newer or current model.

Of current models, even the entry-level iPad8 has specifications and capabilities that will massively outperform you current iPad - and is compatible with the vast majority of current Apps. Other models are more powerful still.

I hope this information provides helpful and succinct clarity as to your available options - and the limitations of your current iPad.