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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.

That last version of Internet Explorer for Mac was released in 2003 & won't run on a modern OS.

The closest you can achieve without actually installing Windows would be to change the User Agent in Safari to pretend to be a Windows native browser.

From iMore - How to view websites on your Mac that require Internet Explorer (or a PC)

How to access websites that require a PC or Internet Explorer

1. Launch Safari.


2. Click Safari in the menu bar at the top of your screen.


3. Click Preferences.


4. Click on the Advanced tab.


5. Check the "Show Develop menu in menu bar" setting, then close the Preferences window. The Develop menu should now show in your menu bar.


6. Go to Develop > User Agent.


7. If you need to pretend you're using Internet Explorer, choose one of the Internet Explorer options.


8. If you need to pretend you're using a PC, choose "Google Chrome — Windows" or "Firefox — Windows".

First, please note that Internet Explorer has been deprecated and replaced by Microsoft Edge for 4 years. It is not known to be a secure browser and has frequent major vulnerabilities. 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
  


• Use a virtualization software (VirtualBox, Parallel Desktop or VMware Fusion) to run Windows inside a virtual machine on macOS


• Use a compatibility layer 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 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 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, 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
  


• 
  CrossOver, 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, 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 :

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).