Is it possible to set up the email notifications for this modules to be HTML emails?

I think so, yes, but I'm not sure how. Drupal 7's way of switching to HTML mails is not easy to understand, and I haven't really looked into it. I think you have to install/code a module that provides HTML mails, then switch to that globally and then you might also be able to send the mails of this module as HTML mails.
But, as said, I'm really not sure of any details, or any blockers you might encounter.

If you find out more, please report back here to help others with this problem!

Hi,

I know this issue is a bit old but it's still active so I'll post what I found here.

I wanted to send HTML emails as well, here's what I found :
Using Mime Mail module (I think it's the only one needed), you get a mail system interface at admin/config/system/mailsystem which allows you to select the default mail type for each module you want.
Just define (full or filtered) HTML mail type for search api saved searches and you're good to go!

regards,
slaurent

@drunkenmonkey : I use a lot of your contributed work in my project so i'd like to thank you a lot for what you did with the search API module(s)!

Thanks a lot for posting this here! I hope it does help someone looking for this information.