The short answer is probably "yes we can", and possibly "we've already seen supernovae from the first galaxies", in the form of long-duration gamma-ray bursts. GRB 090429B has been given a redshift z=9.4, beating the previous record-holder GRB 090423 at z=8.2. As we continue to watch the skies, we're seeing more and more of these objects, and we'll gradually push back the boundary. I don't know what sets the upper limit on how far away we can see GRBs but I don't think it's a coverage problem: Swift covers something like a tenth of the sky.
Note that despite the high redshifts, they're probably not quite high enough to be the first stars. If someone reported a long GRB at z=15, then I'd think it more likely.
I'm not an observer, so I'm not sure about what colours and magnitudes are usually associated with supernova, but I can try. With a bit of Googling, Daniel Kasen's page suggests that they're relatively bright in most bands. Off the top of my head, I think we see them most in optical, but that might just be a selection effect. I think, until now, we've been finding Type Ia supernova up to about z=1.5. That boundary is being pushed, but I'm not sure how. (Possibly improved IR spectroscopy from a Hubble servicing?) The overall brightness of supernovae is of the order of 10$^{51}$ ergs. Type Ia's have typical absolute visual magnitude -19.3, according to Wiki.
As for other types of supernova, I suspect that as you move further out, isolating a single supernova in a low-resolution galaxy image is a major obstacle, but that's pure speculation on my part. That is, the supernova needs to be significantly brighter than the surrounding galaxy. Fortunately, I think this is the case for GRBs, but I'm doubt it for other supernova.
I'm much less knowledgable on this than my previous answer, so I welcome corrections.