You will need to download & install macOS 10.11 El Capitan since apparently Lion is too outdated to access the App Store. If you have 2FA enabled on the AppleID, then you would need to enter the code along with the AppleID password.
You can use any other computer & OS which can access the Internet to download the macOS 10.11 installer since it is available outside of the App Store. You will need to transfer the downloaded .dmg file to the 2013 Mac where you will need to run the .pkg file after mounting the .dmg file. That will extract the real macOS 10.11 installer into the Applications folder where you will need to manually launch the installer. Here is an Apple article with instructions for downloading & extracting this installer....read it carefully:
How to download and install macOS - Apple Support
I highly recommend you create a bootable macOS USB installer so if something goes wrong during the upgrade process, then you will be able to start over by installing macOS 10.11 instead which will save you some time & steps.
Create a bootable installer - Apple Support
Or you can try booting into Recovery Mode using Command + Option + R. If macOS 10.12.4+ was ever installed onto this laptop, then it should boot into the macOS 10.15 or 11.x online installer depending if you have the Early or Late 2013 model respectively. Unfortunately some Macs may only boot to the older online installer for the version of macOS which originally shipped with the computer from the factory regardless of the keys used for booting.