Macbook Pro 11,1 2014 running OS X Maverick, Big Sur not showing up as an update

If that is a 4GB RAM 2014 MacBook Air, the last thing you want to do is upgrade it to Big Sur as the operating system is huge (compared to Mavericks) and will have all it can do to run in that RAM limitation and underpowered CPU. I have first-hand experience there. Whoever had this Mac before you should have wiped it and shut it down so that you will have a new setup screen when it is first powered up. That eliminates the pollution of someone else's Apple ID and software installed on it.

Apple only keeps the current operating system available when launching the Mac App Store from Mavericks so that dog won't hunt.

If you have made a Time Machine backup of Mavericks then you can get your choice of Catalina or (woefully Big Sur) from How to download and install macOS - Apple Support and I recommend using the Safari browser for this effort. Neither Catalina nor Big Sur are supported by Apple any longer and Microsoft has also dropped support for both operating systems.

Understand that Catalina is the first Apple operating system that requires 64-bit applications and peripheral drivers. Mavericks, High Sierra, and Mojave could support 32-bit and 64-bit applications and drivers.

You can download the macOS 10.11 El Capitan installer from outside of the App Store using any computer & OS which can access the Internet. Just transfer the downloaded DMG file to this old laptop & extract the real installer, then run the full installer from the Applications folder. Once El Capitan has been installed, then you should have access to the App Store to download the later macOS installers if you wish.

Here is an Apple article with links to various macOS installers.....make sure to carefully read the instructions since there are multiple steps involved for macOS 10.11 El Capitan:

How to download and install macOS - Apple Support

To use MacOS 10.9 Mavericks for ANYTHING you must FIRST apply all updates using

Software Update > Other ...

... until none remain un-applied.

At that point your should see Safari v9 installed, which is the oldest version of Safari that can make ANY httpS: encrypted connections to anything, including the Mac App Store and the Apple Software Update server.

The historical recommendation seen here is that MacOS 10.11 El Capitan and later requires MORE than 4 GB RAM. Versions after 10.13 require even MORE, because Metal imaging and preliminary support for Apple File System drive format are added at the point. Catalina and Big Sur are likely "a bridge too far" and will likely make responsiveness so poor, that Mac will seem "unusably slow".