I'm a native English speaker (New York) and have never heard reindeer with an S on the end.
Many sources say "plural same or reindeers", e.g. here http://www.oxforddictionaries.com/definition/english/reindeer
We've changed the main translation to reindeer.
Though I can see why you would do that if reindeer is used more often for plural, I still disagree as the -s on the end emphasises that the reindeers is plural. Whilst learning questions, one of the phrases "vart går du?" (where are you going?) explicitly has "where are you going to?" as the main translation to emphasise the point, despite English rarely ending the question with to.
If it helps, imagine that you're using the vowel in the English exclamation of disgust "ugh" when trying to produce the ö sound. It might not be exactly the same sound depending on your dialect, but it should be somewhat close. If your dialect uses a different vowel in "ugh" than mine does, it might not work, but it's worth a try.