Although landing a position working with nonhuman primates tends to be a somewhat competitive venture, volunteer opportunities do exist all over the world for people interested in working in the field of primatology.
To be a competitive applicant, it may be to your benefit to first attend a college course or two in topics such as Animal Behavior, Ecology, Anthropology, and General Biology/Zoology.
Some volunteer positions prefer candidates with degrees and/or previous experience, but you should apply to whatever position seems interesting to you because other factors come into play in the hiring process. Good volunteer candidates can show that they are dedicated, enthusiastic, and willing to take instruction. Be sure to mention any experience you have had with animals, including volunteering at your local veterinary clinic or animal shelter. Oh, and knowing a foreign language never hurts, either, as many of the positions involve work in other countries.
Even without experience or credentials, you can usually find opportunities to volunteer at zoos and wildlife refuges as a docent (someone who educates visitors about the animals) or in other unique areas, such as the enrichment program. Contact your local zoo and request further information on volunteer opportunities. Besides having a fun time, you never know where these types of experiences can lead to in the future!