In theory, the email address of the recipient should be passed back into Vidyard from Eloqua if the video is viewed on that page. The Eloqua system in place should be able to capture the current browser cookie (if Eloqua scripts are present) and associate it with the email address after the recipient clicks. Vidyard can then pull back the email address from Eloqua through the integration.
As a fallback, it's worth adding a vyemail string to the URL of the page where the Vidyard player is located, so that the Vidyard system does not have to rely on this completely.
When looking at the details on the article Associating your contacts using vyemail, you see that HTML code needs to be added to the email template, but the view in Eloqua only shows a "WYSIWYG" or plain-style text editor, as such the one below.
Eloqua has two methods of creating a email template - either HTML or Design.
It's likely you are in the Design version where there is no HTML editor to the right side.
It's still possible to use the Visual editor to add a vyemail string.
To do so, insert the image of your choice and when selected, click Hyperlink from the left menu. When you see the dialog below, click Enable as Hyperlink and then paste the URL.
Once that URL has been added, use the span tags to create the email address merge.
In full, the vyemail string at the end of the URL will look like:
?vyemail=<span class=eloquaemail >EmailAddress</span>
This is making the assumption that you have the Merge Field for 'EmailAddress' available to use (it should be a default Merge Field for your instance).