Is it possible to have edgeR output contigwise variance? I'm looking to determine within and between group variance. 

Honestly, I don't think that you're looking for the right thing. Looking for with and between variances from edgeR is a bit like looking for propellers on a jet plane.

First, variances are only appropriate for normally distributed observations. Variances are not an appropriate measure of variability for RNA-seq counts.

Second, within and between group variances are only a means to an end, not an end in themselves. What you ultimately need to do is assess differential expression, and edgeR provides a wealth of ways to do that. It would be better to use edgeR on its own terms, rather than trying to repeat a traditional analysis that is not appropriate for this type of data.

edgeR will analyze whatever counts you give it. Most typically these are counts for a gene across all samples, but they can also be counts for genomic regions, transcripts, etc. Supplying counts for a contig will work just fine; in which case, the tagwise NB dispersion corresponds to the "within group variance" for each contig. I presume you got the "between group variance" concept from ANOVA in linear models; the equivalent is ANODEV in GLMs, which is described somewhere in the edgeR user's guide.