I now have two Public Lab fold-up spectrometers working with iPhones. These can take still photos of spectra to be uploaded to SpectralWorkbench.org for analysis, but there is not yet an app that streams video to SpectralWorkbench. The still photos are probably better quality than the video, so that's fine with me.
I made a couple of modifications to the spectrometers. First, I used grating replica polyacetate film instead of a piece of DVD because I have lots of it and it appears to transmit more light (makes a brighter spectrum). This required that the angle of the spectrometer be changed by 3 degrees, which I assume is not necessary if a DVD shard is used [Update: I now think the angle has to be increased by more than 3 degrees]. Second, I made the entrance slit narrower by about 50%. This is easy to do, and allows spectral peaks to be resolved more precisely, but might not be important for many applications.
The video below documents the assembly and adjustments so far. This research note describes the first field test of the spectrometers.