Trying to get research experience in a PsyD program is generally going to be an uphill battle, as clinical work is typically the focus. Having said that, there are excellent PsyD programs out there where doing research would be an expectation. These are also the most competitive PsyD programs and tend to have funding as well (i.e. Rutgers, Baylor).
I have a few friends with PsyDs who were able to get some research experience in their program. However, they were incredibly self-motivated and pursued research opportunities with the faculty, putting in work beyond their time spent with classes and seeing patients. So I suppose the answer to your question is technically a yes.
Having said that, I encourage you to consider PhD programs instead of PsyD programs - for 2 primary reasons. First, the majority of PsyD programs are not funded and you would be looking at a minimum of >$75,000 debt for the entire program. PhD programs typically are funded, on the other hand. Secondly, and perhaps more importantly, you stated that your goal is to primarily see clients but also conduct research on the side. Most clinical PhD programs will provide in-depth research training but will also provide great clinical training. On the other hand, PsyDs will have a much weaker research component (if any), and generally less opportunities for you to gain that kind of experience and develop competency.
I believe there's a statistic that like >70% of people who graduate with clinical PhDs end up becoming clinicians rather than going into research careers. Even if that is your end-goal, there are PhD programs that are more balanced, where you will get good clinical and research training. I urge you to consider these things as you go forward.