Let's say that we have a perfect algorithm for imputing data.
You give him a dataset with missing data for some features and it predicts them.
I had had such an data imputation algorithm, I would have use it as a classifier.
The fact the we can reduce classification to data imputation means that dat imputation is as hard as classification.
As Rayn said, you can indeed do so.
Imputation algorithm could have been evaluated as classification algorithm but this is not very common. One of the reason is that the rule by which you hide data from the imputation algorithm is subjective and might effect the results significantly.
In whatever method you use, you assume that the data behave in someway and missing due to some rule.
It is also an important question, since errors in the imputation (you will probably have such), will effect the process down the road.
I prefer to avoid imputation and let the prediction algorithm cope with the missing data. There are many such algorithms.
A different approach is to use few methods of imputation, choose few simple classifiers (so their complexity won't have a too large influence on the results) and compare the predications.
In many cases, there is no big difference.
When there is a big difference, of course that you will prefer to proceed into a more detailed analysis using the winner.
However, before that try to understand why it has a big advantage. That might lead to interesting insights about your data.