According to this:

https://superuser.com/questions/1101623/what-happens-when-you-go-to-sleep-mode-in-windows-10/1101647

…sleep stops the cpu from running, so no programs run while Windows is sleeping.

Do you mean sleep, as such, or maybe instead what happens when the screensaver cuts in?

Hi Zenos,

Thank you for getting back to us.

When screensaver cuts in, will this prevent receiving emails automatically?

We are using Thunderbird on our clients computer to automatically print certain emails. Sometimes they do not receive any emails. The only way they receive it when they refresh Thunderbird.

How can Thunderbird check for new emails, even when the screen saver is on?

thank you

As long as the OS is actually running and not sleeping, then thunderbird is running and receiving mail. How/why it does so isn't very relevant