Do sites using YouTube video API get indexed by Google? The answer is using these API's will not prevent the site being indexed.
...what options do I have for SEO?
Google will read rendered code. So it does depend on how the code renders when the bot reads your site.
However, if the API translates to just embed or object (or similar controls) then there is nothing (very little) for Google to read (although my research suggests it does of course understand it, but this may not add much value).
Again, it depends on what video's you're serving and why to answer what options you have with SEO but, and assuming the videos are not random, then you could have pre-saved information about each video link. This way, based upon what video is pulled back you then load the appropriate content. This will give Google something to read. However, if the page is constantly loading new videos and therefore new content, Google will not know what that page actually is about since every time it visits, the content has changed completely.
You could take the view of categorising the page or consider having multiple pages for this. So, and assuming you have some control here over what video's you're pulling across, then have some generic content on the page for Animals and only pull animal videos. Have generic content for Music and only pull music videos etc.
Now, I think combine the two! Make sure you have static content on the page which explains the videos at an abstract level which will remain constant and then focus on 'article' (video) detail in a smaller way.
Of course, all of the above is fairly useless if the site is never indexed so you will need to submit the site to Google.