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I need to connect to the website dictate.it, which says it only works with Internet Explorer.

Is it possible to find versions of IE to run somehow within macOS?

You can use a virtual machine and then use the images provided by Microsoft for use in testing older versions of IE. Requires no cost and you can delete it when you've finished using the site.

For free you can use VirtualBox, and then use one of the images from Microsoft.

An alternative to virtualization (which is already covered in other answer) would be to use a Windows runtime environment emulation such as Wine (free & open-source).

PlayOnMac, the Mac version of PlayOnLinux, is a free graphical frontend for Wine and provides configuration templates to run the windows version of Internet Explorer.

Alternatively, you can also install and setup Wine manually. It's mostly dedicated for advanced system administrator but there is a packaged commercial front-end called CrossOver with customer support if you have some money to spend.

Although those solutions have a lot of benefits (no virtual machine, no Windows license needed, 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 be the easier way to go.

Edit: I managed to run Internet Explorer 8 on macOS Mojave using the trial version of CrossOver :

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.

You could try to use some of the cloud tools that provide access to browsers via their service.

Essentially you create an account with one of these services, and through their dashboard you can access a list of browsers that you can control through your own browser.

One such example would be BrowserStack who have partnered with Microsoft to allow developers to test freely on Edge. They have a free plan which will allow you to access Internet Explorer and get your job done.

There are other providers too that you could consider: Sauce Labs, Cross Browser Testing, Browser Ling

Disclaimer: I used to be a BrowserStack employee

You have a few options:

• 
  VM software and an IEDev image. Microsoft provides free (time-limited) versions of Windows with IE in the form of virtual machine images. You can get them here: https://developer.microsoft.com/en-us/microsoft-edge/tools/vms/
  


• 
  Install IE using WINE. The WINE project is a Windows compatibility layer for Unix-like systems including macOS. It's free but installing IE can be a pain. You can use tools like Winetricks to make installation easier. Good if you need an earlier version of IE than Microsoft's official solutions provide.


• 
  Change your useragent. In many cases, it used to be that most web browsers would work fine (or at least be operable) on sites optimised for IE only. However, developers couldn't be bothered to support anything outside of IE and would simply put a useragent-based whitelist in place. If you spoof your useragent to be IE, you might still be able to use the site. You can find many extensions for this in Firefox and Chrome/Chromium.

Buy Parallels, buy Windows, install Parallels, install Windows, and now you can run Windows applications inside MacOSX, including Windows internals like Internet Explorer if I'm not mistaken.

Costs a few hundred dollars total, but you now have Windows running inside MacOSX and can run most Windows software directly. Some exceptions exist, especially late versions of DirectX and OpenGL aren't supported because of restrictions imposed by OSX (Apple's stonewalling on providing an up to date OpenGL implementation).