1) I recommend everyone I talk to, regardless of their undergrad prep, to do SOME kind of volunteer counseling work if possible (echoing what CheetahGirl and erg923 have said). Not so much because it is crucial to your applications, but because you should really find out if you like or hate the work before you dedicate years of your life and lots of money to it! I also volunteered as a rape crisis counselor and advocate for two years - it was incredibly hard, but I loved the work, and I had others who started training with me who were also interested in psychology jobs but who fizzled out because they couldn't stop taking clients' problems home with them. Good to learn this early.
2) Also echoing what erg923 said - you may be able to pursue your career goals with a master's level credential, which won't place as much emphasis on research experience. A great resource from Mitch Prinstein at UNC goes over the different kinds of psychology-related degrees (PhD, PsyD, MSW, LMFT) in a summary way, and then you can research more online for degrees that seem to fit what you want:
http://mitch.web.unc.edu/files/2013/10/MitchGradSchoolAdvice.pdf
3) Are you near a major metro area and/or an area with universities? Connecting with a program face-to-face is going to help a lot with research experience and eventual letters of reference. Look up the psychology departments of programs near you, find faculty doing research you're interested in, and email them directly to see if they are looking for volunteer research assistants. (Sometimes you can get paid gigs, but you're better off starting looking for a volunteer position that you might be able at some point to transition to paid, rather than looking for paid directly. Of course, if you find a paid position that's a good fit for you, go ahead and apply! Just don't limit yourself to only paid positions). I was a lab manager for two years and while we primarily had undergrad RAs from that school working in the lab for course credit, we did have funding to pay outside volunteer RAs if they were really great and we wanted to keep them on after they volunteered and proved they were reliable, etc. for a term or two.
4) Major psychiatric hospitals will often have ongoing research projects, either externally funded or internal, program evaluation-type research. You can look up research projects that have federal funding through the NIH reporter system:
https://projectreporter.nih.gov/reporter.cfm (note you can limit your search by city and state). If something seems reasonable and interesting to you, get in touch with the project manager or PI (google is your friend here) about volunteering. Keep in mind, generally speaking, if you don't have experience working with clinical populations (E.g., people in the hospital with psychiatric issues) you probably won't start any position working with patients/clients directly; typically volunteers sort of "work their way up" from filing, photocopying, assembling study materials, entering data, to running subjects through studies and then eventually being able to look at and analyze data yourself (again, this will vary somewhat by lab).
5) To the extent that you can, try to really engage with the professors in your online classes. You'll need three letters of reference for grad school applications, and it may be hard to get all three from a research environment via volunteering. It'll also be hard to get an online class professor to write you a good letter if all they know about you is "student number 123456 got an A in my class." Ask questions, be engaged, try to connect with the professor in some way so you stand out. Also, if there's any chance that you could physically meet with any of the online professors (or even volunteer in their labs - if your school is physically located somewhere near you) that's not a bad idea.
6) I know less about research-focused master's programs, but I had a friend from my previous lab who ended up doing such a program at Wake Forest University, and another friend who did her undergrad there and spoke highly of the program. Otherwise, your best option is to look online to find programs that are master's level and read closely about what types of training they provide (e.g., how much research experience do you get, what kinds of outcomes do their students have in terms of moving on to PhDs or getting the jobs they want, etc.).
Good luck, OP!
Edited to add: Yes, I think there's stigma around online-only programs and the type of education they can provide, although I think there's also stigma about other kinds of programs as well (e.g., schools known for grade inflation, not-flagship state schools, etc.). The best thing you can do at this point is to make sure the rest of your application is as strong as possible, particularly things like research and your letters of reference.
