Unfortunately not. Apps are tied to an email address, and for Chrome OS this would be the user that logs in. Supervised users work differently. They're not tied to an email address and so they cannot install apps. Basically, the two features (supervised users and chrome apps) are built on foundations that don't coincide very well.
The Chrome OS developers are aware of this problem and that many people are interested in this feature, as noted within this Chromium Bug Report
My children also have Chromebooks with supervised user accounts. We maintain a whitelist of sites that they can visit at any time without parental supervision. The problem with this approach, in addition to not being able to install apps, is that the "discovery" factor is destroyed - they don't have the ability to find new sites of interest.
What we've recently done as a workaround is to have monitored time where they are signed in as non-supervised users. On those accounts, we've installed AdBlock Plus, which removes a lot of bad content. Additionally, our router now assigns DNS using the free OpenDNS Family Shield. This blocks access to many many websites that we would not want our children to stumble on.
While AdBlock and Family Shield work well, it's not a fool-proof system. There is still much objectionable content on the web, and even that is subjective. This is why we still choose to monitor their usage during these times.