Is it possible to run Internet Explorer on OS X El Capitan? 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.