Starting with a couple of reasonable assumptions, that the mixers you are describing are relatively small, impeller diameters of less than 18 inches in diameter, and that the rotational speeds are relatively high, greater than 350 rpm.  The specifics of impeller size and rotational speed will give different answers to this question.  Although options also exist for larger mixers.
 
The mixer shaft should not see much erosion (as caused by abrasion) because the surface velocity is relatively low, although corrosion could be a problem.  Most mixers do not have plastic shafts, because the rigidity of plastic is such that the shaft could behave like a wet noodle when rotated.  Some large diameter (greater than 3-inch diameter) mixer shafts have been made of fiberglass reinforced polymer (FRP), but those are typically special purpose, other than possibly available from a custom fabricator or a pipe supplier.  Design of any type of polymer shaft requires a knowledge of shaft stiffness and a natural frequency calculation.
 
Impellers are not typically made of polymer materials either with a few exceptions.  High speed disperser blades are available in either polymer material http://norstoneinc.com/polyblade_intro.html or with hard coating http://www.bladeshop.com/hard-coat.htm.  Other suppliers may carry similar blades.
 
For larger mixers, coating options are often used for abrasive and/or corrosive service.  Sheet material such as natural or synthetic rubber and dipped or sprayed polymer materials are available as coatings.  The specifics and success of coatings depends on the nature of the erosion and the chemical composition or temperature of the process.


The answers by this expert are based on the best available interpretation of the information provided.  The consequences of the application of this information are the responsibility of the user.  If clarification is needed, please submit a further question.