The correct term is ‘sport psychologist’ using the non-plural version of the word ‘sport’.
One of the reasons why the term ‘sports psychologist’ (technically incorrect) gets used almost as much as the correct term ‘sport psychologist’ is due to two reasons. First, they sound exactly the same when you say them (try it). Secondly, from a logical point of view if the psychologist works across many sports (as opposed to just one – which most of us do) then it might make more sense to use the plural version of the word sport.
For more on this subject read this very informative blog post by Canadian sports psychologist Kate F. Hays where she correctly points out that the original correct spelling was actually without the s – so sport psychology and a