A few things to add to what others have already said:
-Missing data in factor analysis is a royal pain for a number of reasons. A friend of mine (Ed Merkle) wrote his dissertation (at Ohio State) on the topic. You may want to get in touch with him: http://psychology.wichita.edu/people/Faculty/Merkle.htm
-Multiple imputation in the context of Exploratory Factor Analysis is liable to run into the issue of label-switching. In factor analysis, the signs of loadings are not identified and the iterative solution may well flip the signs around quite arbitrarily. CFA would be much more tolerant because you are imposing much more structure on the solution and thus will constrain things enough to avoid label-switching.
-(Stata's EFA program has some issues, most notably lack of standard errors, inability to use a covariance matrix and a very restricted number of algorithms, most notably lacking GLS and Minres/ULS; it also does sequential chi square tests for dimensionality selection which are not statistically valid. On the other hand it has some really nice features, such as the ability to do multiple random starts.)
-You may consider looking at software that handles such things, such as Mplus or AMOS, which will do missing data analysis and factor analysis all at once. I'm not familiar enough with GLLAMM to know how it's going to handle the missing information, but you could certainly set up a full information ML factor analysis in it. Warning: It will likely be VERY SLOW.
Jay
PS: Hi from a former Illini. :)
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