Updating iPad 3 to latest software?

No.

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 need to consider that compatible Apps for your iPad are increasingly difficult to find - as App Developers have dropped all support for 32-bit devices and many for iOS versions 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. As such, if you want or need to access most current Apps, requiring more recent versions of iOS than you iPad is capable, you’ll need to consider replacement of your iPad with a newer or current model.

Of current models, even the entry-level iPad9 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 succinct clarity as to your available options - and the fundamental limitations of your current iPad.

A 2012 iPad 3rd generation is over 10-years old, now, in 2022.

That iPad can never, ever be upgraded/updated beyond iOS 9.3.5/9.3.6.

That year and model iPad is, simply too old and it is, also, obsolete.

Sorry.

Wi-Fi only model: iOS 9.3.5, released August 25, 2016 

Wi-Fi + Cellular model: iOS 9.3.6, released July 22, 2019

those are the most up to date versions supported on your iPad 3