Teaching with a Psy.D. is definitely possible, but there are a lot of barriers as others have mentioned. I'd say your best bet is to teach within a Psy.D. program itself, followed by some of the academic positions that are a little lower on the prestige scale. That said, there are always exceptions and if you are a prolific researcher and teacher, then the sky is the limit. But if teaching is a primary goal, I wouldn't usually recommend the Psy.D. unless all you plan to do is adjunct in addition to your primary clinical job. Ollie mentioned some other possible settings as well. But the trend these days on the academic market - even at SLACs and community colleges - is to bring in people who know how to conduct research (so that they can get undergrads involved in their studies). A lot of Psy.D. programs don't teach students the skills to actually conduct research studies, which I'd argue is one of the biggest distinctions between the degrees. But there are some Psy.D. programs that do a better job of this.
So, I think you can make it work on some level, but the deck is stacked against you and there is a stigma against Psy.D.s in some settings. You can overcome it if you are really good at what you do and network very effectively.