I think emailing the gym is the best way. Introduce yourself (along with any credentials) and present your project. Inform them of everything you would need from the gym as well as the timeline for completion. End with your contact information and wait for a response. Keep it professional and be patient. If they don't respond within a week, I think calling is okay. Ask to speak to the gym owner/director. But, if you are denied twice, then leave it alone and try somewhere else.
I know that this method is direct, but I think that's the best way to get someone's attention in a professional manner.
You can also try reaching them on social media and inviting them to join a challenge by posting on their page.