Would it be safe to install Internet Explorer on my iMac?

Internet Explorer has reached end-of-life status with Microsoft. It is no longer included Windows10. There is no option for Internet Explorer on the Mac.

The website your wife uses is the problem. If Internet Explorer is necessary and no other option is viable, I have every version of Windows from Windows XP all the way to Windows 10 loaded on virtual machines and using VMware Fusion, I occasionally use whatever version (usually Windows 7) to check certain browsers, including Internet Explorer.

Internet Explorer is Not Compatible nor available to Install on macOS Big Sur or for that matter on macOS of almost any version. As Colleague @Ronasara has mentioned there are Others available just be sure it will function on the new Big Sur

Internet Explorer is a older browser no longer supported by either Apple or Microsoft. Typically today, the best options are Google Chrome, which can be found at www.google.com/chrome or Firefox which can be found at www.firefox.com Both of these are free to use and allow for full compatibility across platforms of both Microsoft and Mac.

There is also a option under Safari that allows for Microsoft emulation. It allows Safari to mask as a Microsoft browser. https://www.imore.com/how-view-websites-your-mac-require-internet-explorer-or-pc

Add to list of possible Browsers - MS Edge - have not used on macOS so . . can not offer first-hand experience.

Yes. In fact it can be useful to have multiple browsers for various reasons. Some web sites will only work with specific versions of specific browsers.