I have been trying to figure out how to use HTML in pop-up windows (ArcGIS.com viewer). It sounds certain that it is not possible with ArcGIS Explorer Online. The next option is the ArcGIS.com viewer.
Someone from ESRI stated that HTML could be placed in attribute columns of a feature class that is being served by ArcGIS Server. The pop-up window could use the HTML. This will be messy and difficult to maintain. For a simple bit of html it might be ok.
I have been using html in some of my maps. I don't know if that is any help to you, but I thought I would share it.
If you click on a firetruck symbol in this map you will see a picture in the pop up window and are also able to click on it and it will take you to a website.
Here is the code I used: <a href="Website URL" target="_blank"> <img src="Your Image URL" /> </a>
I just create a field in my shapefile where I put this code and in the pop-up window configuration used that field.
I have also been playing around with putting videos in my pop-up. It works fine in the firefox browser, but there are problems with the chrome broswer.
Thanks for the feedback! The fire map works fine, the problem with the video map is only when using google chrome. When you open the pop-up window and play the video clip and close the popup window the video will keep playing. I talked to ESRI about it and they try to fix it.