How can one find a research mentor to fulfill a research paper requirement prior to PhD application?
Considerations about the paper


Before anything else, I would suggest checking with that program about what they consider a research paper. It is possible that something like a literature review, or a research proposal, rather than an original research study, would be adequate for their purposes. 


If they mainly want to use this to gauge your writing ability or your familiarity with the scholarly literature, then a literature review should give them most of the information they need. 
If they want to see your understanding of experiment design, then a research proposal would help give them that, on top of containing a literature review section.

Do you have any past coursework that has produced anything remotely like a research paper? If so, starting from there and revising would be a good start.
To actually conduct a research study, you will need to either reanalyze publicly available data or gather new research data.


If you're reanalyzing data, you might do some sort of statistical project, or a qualitative project using, say, library archives of old or anonymized interview or case data.
If you're gathering new research data, you need to make sure you are getting informed consent and using confidential data correctly. This might be very challenging outside of a university and/or hospital Institutional Review Board setting, so a research mentor would be critical here. (While class projects and pilot projects are usually exempt from such review, you want to make sure that nothing you do appears to cross ethical lines.)



Approaches to finding a research mentor to work with


Through your work


Do any of your colleagues at/through the nonprofit have a Psy.D. or a Ph.D.? They may be able to help you with this.
Do any board members of the non-profit have academic affiliations or doctorates? They, too, may be good to approach, either as research mentors themselves or as people who could connect you to appropriate people.

Through your alma mater


Even if you have not been in close contact with professors from college (and/or your Master's program, if applicable), you can still ask them for advice.
If you are far from those colleges, you can also try to do some "cold-calling" through the alumni network, if there is one with information available to you. (If there is no official one, you could try through LinkedIn, looking for people with a Ph.D. or Psy.D. in a related field in your geographic area.)

Through nearby colleges


Is there any summer psychology course at a community college, college, or university nearby that involves producing a paper? Check with the faculty there and see if there's a way you could get their guidance on a research study through that course.
Faculty at nearby colleges who are actively conducting research (see if their website/online profile mentions "research") may be willing to help you with this in exchange for, say, you helping conduct interviews for their research. You could reach out to the associated departments at the colleges, offering research assistance in return for some mentorship (and the department will share info with the professors), along with perhaps a few personalized emails to professors whose work would be a very good fit with your own.

People in other fields


The more "generic" parts of writing a research paper, such as structure, doing a literature review, and tone, don't require someone who is specifically in counseling/psychology to help, and getting others to help edit your work can improve it immensely.
See if your local librarians can help you get access to the research literature, and/or recommend ways that you can gain access to a local college library. They may also be able to help you with the basics of approaching a bibliography or literature review.

Through the program you're applying to


If you've talked with the program specifically about this requirement, perhaps you could volunteer to work with one of those professors to help conduct research and/or analyze research data this summer.
This would take some chutzpah but would potentially get you a strong advocate for your admission, if done right.
At the same time, faculty may be leery of (remotely) mentoring someone new at research, before the admissions decision. Plus they might have other plans for the summer.



In short, you'll be more likely to be able to do a literature review or research proposal rather than a research paper on this time frame and without university resources (human subjects procedures, etc.). However, depending on how widely you reach out, you may find people who are able to mentor you through this.  Good luck!