As a Reservist NCO on a deployment as a contractor, how should I go about my next NCOER?

No, you may not put any of your contractor duties or accolades on your NCOER.

I worked for many years as a government contractor and had team members who were also part of the Reserve Component. The military and public sector jobs must remain separate, but can be complimentary. Your military evaluation, NCOER, must be written on your performance as a military member, that is, the time you spend at drills, temporary active duty, and other official military duties. I'm not sure of the details, but some military professional development classes may provide drill time or points (check on this with your chain of command) and might be included in your NCOER as a comment about self-improvement.
Your public sector job should be evaluated separately from your military job. Your supervisor will follow corporate policy and guidance with regard to writing an evaluation and conducting evaluation counselling. It's likely any specialized training you get in the military may be mentioned in your civilian evaluation as efforts at self improvement. Overall the quality of your performance as documented in your civilian evaluation should greatly influence your future compensation and continued employment.
Keep your performance goals for each job separate in your mind and strive daily to exceed the quality standards in both of your jobs and you'll be fine.

The purpose of the NCOER is to evaluate your time as an NCO in the position you are assigned. Nothing you do outside of those constraints can be evaluated by your rater. The exception would be if you are adcon to another unit, and your supervisor can provide a LOI. But that is still subject to the whims of the rater.
If you are deployed as a contractor, and not drilling, that time would be coded as unrated time, as MAJ Javier Rivera mentioned.
If you are still drilling, only the time spend in UTA or AT status can be evaluated by your rater. ADT status is also unrated time as you would get an AER for that specific timeframe, so any time spent in training would be unrated on your NCOER, but you get a 1059 anyway. Civilian recognition is tricky, in some cases it may be appropriate to mention it, but it is very specific and must relate to the accomplishment of duties performed while in the appropriate rated status.