Most of the time, you can re-download the current version of macOS via the Mac App Store, and older ones via these links:

1. Download Macos High Sierra Full Installer
2. Download Macos High Sierra Full
3. Macos High Sierra Download Offline Installer

However, I’ve run into a situation several times where the Software Update mechanism simply refuses to initiate a download:

Thankfully, macOS installers can be downloaded via Terminal in macOS Catalina. This command will download the most recent version of macOS, depositing it in your Applications folder:

How to download High Sierra installer. So it won't be much help if you are trying to get the full download of High Sierra. Ask them to download the Mac OS X installer for you.

1. Download MacOS High Sierra v10.13.6 free standalone offline image installer. MacOS High Sierra 10.13.6 is a powerful OS X release from Apple with numerous enhancements and a bundle of powerful tools.
2. I was able to download the macOS High Sierra 10.13.6 full installer version thanks to a mid 2012 macbook pro with macOS El Capitan installed. All other attempts with the latest mac book pro and already installed versions of the operating system (Catalina or Mojave) have been unsuccessful.
3. Download macOS Sierra This is an official macOS Mojave Sierra from Apple, It can be used to Reinstall or Upgrade to Sierra on a Mac. If you are looking to Install Sierra on PC read Hackintosh Sierra Guide.
4. Download the macOS Sierra installer. Launch the App Store app, then look for macOS Sierra in the store. (Here’s a link.) Click on the Download button, and your Mac will download the installer to.

softwareupdate --fetch-full-installer

The softwareupdate command has some neat tricks up its sleeve, as pointed out by Armin Briegel:

The --fetch-full-installer flag has a sub-flag: --full-installer-version which allows you to download a specific version.

During my testing in the Catalina beta version I was able to download 10.15, 10.14.6, 10.14.5, and 10.13.6. I was not able to test if 10.13.6 would download the hardware specific build of 10.13.6 for the 2018 MacBook Pro, since I do not have that hardware.

Download Macos High Sierra Full Installer

So, to pull 10.13.6 down, you’d use:

softwareupdate --fetch-full-installer --full-installer-version 10.13.6

Download Macos High Sierra Full

I wish Apple would just have a support document up with direct downloads for all of this stuff, but this tool is not a bad alternative.

Macos High Sierra Download Offline Installer

Update: Don’t miss this documentation from JAMF for more on the subject.