iPadOS 15.7.6 will invariably require more system storage than iPadOS 13.x - but consider this:
1) As is you cannot install many Apps that require a higher version of iPadOS than is already installed - limiting usefulness and utility of your existing iPad. If necessary, shedding some less used or incompatible Apps may make your iPad more useful than is currently the case.
2) iPadOS 15.7.6 includes vital system security updates for vulnerabilities that are being actively exploited. Update is strongly advised if your privacy and security - and that of your personal and sensitive data - are important to you.
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
Be aware that while an iPad may be capable of update, being so far behind the most recent compatible version, you may need to use a computer to perform the update. To do so, you will need access to either a Windows PC (with iTunes for Windows installed) or a Mac (with iTunes or Finder - as appropriate for the installed version of macOS). Ensure that your installed version of iTunes has been updated to the most recent version; the essential device drivers are included.
Download and use iTunes for Windows – Apple Support
Update to the latest version of iTunes – Apple Support
Update software on iPhone, iPad or iPod touch in iTunes on PC - Apple Support
You should note that for your iPad to install an available update, its battery must be charged to at least 50% capacity - and/or be connected to its Power Adapter. Similarly, the iPad can only be updated over a standard WiFi connection - or be updated by a PC or Mac using iTunes or Finder - over a USB connection.
If you have insufficient space to perform the update, you may need to delete unused Apps or unneeded data.
These support pages should prove to be helpful:
About storage on your device and in iCloud – Apple Support
What's the difference between device storage and iCloud storage? – Apple Support
Manage your iCloud storage – Apple Support
How to check the storage on your iPhone, iPad and iPod touch – Apple Support
Clear Other storage on your iPhone or iPad - Apple Support
“Offloading” Apps can be a helpful method of creating temporary space for a software update.
https://9to5mac.com/2020/01/21/how-to-offload-apps-in-ios-to-save-space-without-deleting-their-data/
Additional information about creating sufficient space for installation of a software update can be found here:
If you need more space for an update on your iPhone, iPad, or iPod touch - Apple Support
External storage options can often help with managing limited internal storage. For iPad models with a Lightning port, to use external USB storage, you’ll need an Apple Lightning to USB3 Camera Adapter:
- Lightning to USB 3 Camera Adapter
https://store.apple.com/uk/xc/product/MK0W2ZM/A