Throughout this blog, I've shared a number of videos from A Música Portuguesa a Gostar Della Própria. It's an incredible resource for Portuguese music of all kinds, with over 2000 videos posted so far. I strongly encourage you to investigate the many wonders within.
I was going to suggest searching their site for music from the Azores, but as it turns out the search engine isn't all that helpful unless the search words you use happen to be included in the title of the video. So I've spent the morning digging around there for all of the Azorean music I could find using multiple search terms. I'm sure I probably missed something, but it's a good place to start. I'd post them all here but there are over 50 of them, so it would take forever for this page to load, and I'd rather not do a bunch individual posts.
I thought I'd organize them all into a personal Vimeo channel for you, but after an hour of trying without success it appears that the settings on the videos allow them to be shared in all kinds of ways but not by adding them to a personal channel. Instead, you'll just have to find them in my Likes section, which is easy since they are the only things in there. So go watch them there!