Is it possible to dockerize Internet Explorer?

Docker does not currently work with Windows based applications. The Docker on Azure is Docker running on Linux VM's in Azure. With the advent of Docker for Windows you would think it would be possible, however IE/Edge does not support headless mode. Microsoft has announced that Edge is switching to chromium which will hopefully, finally, bring headless mode to IE/Edge. Docker will never support a GUI environment, it runs counter to their mission.

You can install wine, and IE6 or IE7. I guess you would prefer IE 10 or IE11.

Have a look for example at

https://hub.docker.com/r/tianon/wine/

or

https://hub.docker.com/r/jess/wine/,

and according to winehq.org, you can use an outdated browser such as IE7, if you modify the previous Dockerfiles.

Do not forget you will not have all the DLL available with wine.