Look the link below:http://www.thefreedictionary.com/reindeers
It seems both are possible, however, only "reindeer
" both singular and plural are widely accepted. The vast majority will use the same word (with no "s").
I could find some explanation of a few words in English that do not change from singular to plural, usually it happens due to the origing of the word. Although these words are quite old, they have not changed particularly in this case: singular/plural.
Other examples: sheep/sheep, moose/moose and fish/fish*
All animals, funny, inst't it?
* this last is polemic, it's worth seeing...
better to go with the flow. So, following the usual form, your first sentence
would be my choice.