I agree that it sounds like an oxymoron, as the original purpose of the Psy.D. was to train students for clinical practice. "Sports Psychology" is a very broad term and touches on a number of different areas. Traditionally it includes aspects of positive psychology, mindfulness, attention/focus, etc. While some may argue that parts of it are not "clinical" in nature, it would be ethically questionable for someone to work in this area without being licensed.
I only know of a few sports psychologists, and they all came from traditional areas and only later augmented their work to fit into "sports psychology". The more therapy-centric work within sports psychology involves more traditional avenues suck as body image, self-esteem, identity, managing the pressures of being student "pro". Some people work with athletes on a team basis, and that work seems to straddle clinical/non-clinical work....but again I wouldn't recommend it without licensure.
The University of Oklahoma has an internship spot (and maybe post-doc?) in sports psychology, where the clinician in training works with one of their sports teams. The placement covers a range of opportunities, though it is highly competitive and one of the few of its kind in the nation. The applicants that I met looking at the placement came from a range of programs, but they all studied in relevant areas. It wasn't my cup of tea, but it looked like a great opportunity for anyone looking to work in the area.