First, please note that Internet Explorer has been deprecated and replaced by Microsoft Edge since 2015. It is not known to be a secure browser and has [frequent major vulnerabilities][1]. If your service provider doesn't support any modern cross-platform browser (such as Mozilla Firefox or Google Chrome), **it may be time for a change of provider or service**.

That being said, if you really must use Internet Explorer and since it's only available on Windows OS you only have three choices :

 - Use a Windows PC or install Windows on your Mac using [Bootcamp][2]
 - Use a virtualization software (VirtualBox, Parallel Desktop or VMware Fusion) to run Windows inside a virtual machine on macOS
 - Use a [compatibility layer][3] to run windows programs on macOS without virtualization

That last choice may be a good alternative if you only want to run one specific program and don't need a full-fledged Windows running. [Wine][4] is a free and open-source solution capable of running windows program on Unix-like systems (Linux or macOS).

You can [install and setup Wine manually][5] but it's mostly dedicated for advanced system administrators. However, there are multiple packaged front-ends for Wine that allow running windows program on macOS without much technical knowledge :

 - [PlayOnMac][6], the Mac version of PlayOnLinux, is a free software and provides a user-friendly graphical interface with configuration templates to run the windows version of [Internet Explorer][7]
 - [CrossOver][8], a commercial solution with customer support if you have some money to spend. It also provides a graphical interface and may have better support than PlayOnMac.
 - [Winetricks][9], another open-source solution but mostly dedicated to advanced users

Although those solutions have a lot of benefits (no virtual machine, no Windows license needed, possibly no cost), they may be really tricky to set up if it doesn't work out-of-the-box. In which case a windows virtual machine may still be the easier way to go.

FYI, **I managed to run Internet Explorer 8 on macOS Mojave** using the [trial version of CrossOver][10] :
[![Internet Explorer 8 running natively on macOS Mojave using wine through CrossOver front-end][11]][11]

It may very well be the best solution if PlayOnMac doesn't work and you don't have the technical background necessary to setup Wine manually and you don't want to spend ~$200 on a Parallel Desktop + Windows 10 VM solution.


  [1]: https://www.cvedetails.com/vulnerability-list/vendor_id-26/product_id-9900/Microsoft-Internet-Explorer.html
  [2]: https://support.apple.com/boot-camp
  [3]: https://en.wikipedia.org/wiki/Compatibility_layer
  [4]: https://www.winehq.org
  [5]: https://wiki.winehq.org/MacOS
  [6]: https://www.playonmac.com/en/
  [7]: https://www.playonmac.com/en/app-531-Internet_Explorer_8.html
  [8]: https://www.codeweavers.com/products/crossover-mac
  [9]: https://github.com/Winetricks/winetricks
  [10]: https://www.codeweavers.com/products/crossover-mac/download
  [11]: https://i.sstatic.net/SOnBt.png