Picasa was killed by Google in 2016. The replacement, Google Photos, seems to have a developer UI, but I'm afraid I haven't taken a close look at it to see if it can be supported by the Chumby. The biggest issue would be how to authenticate.
Flickr is what Flickr is.
Photobucket was actually the hosting service of choice for Chumby, but then they changed into a rather expensive subscription service (I think to integrate with a third party device like a chumby would require the user to get a $399/year subscription).
Instagram was acquired by Facebook, and Facebook effectively rendered all of their third-party APIs useless for sharing content outside of their website and branded applications.
All-in-all, it's a pretty poor landscape for hosted photos nowadays compared to just a few years ago - again, another symptom of the steady closing of the web.
Open content syndication like RSS is mostly gone, in favor of closed systems like Twitter and Facebook.
I'm basically sitting on a design for a new, very updated Chumby device, but given the state of the web, it's not clear where the content people would want to see would come from without cutting complicated business deals, assuming that's even possible now. As I think I've pointed out before, even *Microsoft* couldn't get broad content support for their mobile devices - for instance, Facebook, Messenger and Instagram branded apps were pulled a couple of months ago.