OS updates on a MacBook without Apple ID
Yes and no. Upgrades can require an ID, updates do not.
With an MDM, you can push out most apps and updates without any Apple ID needed on the machines. Also in the yes, many but not all OS updates are pushed from the device channel and not the user channel of the App Store.
No for some cases if the machine is older. It really depends on exactly which Mac you have and which specific build of OS you want to install and if you install it from an already set up user account or if you install from recovery.

https://support.apple.com/en-us/HT204904

The easiest way to check is to go to this page in Safari browser and then try to download whichever OS installer you want.

https://support.apple.com/en-us/HT211683

Another less easy way to test is to use recovery and reinstall the OS on top of your existing one. You will quickly know if that machine needs an Apple ID (less likely) or if it’s in the clear (more likely). You only lose some time with this test as the programs and data remain x especially if you pick to reinstall the same OS version and do not upgrade.