Post signs in your local community center and other physical locations where people usually visit. Ask your friends if they need a dog walker. Ask them if they know anyone who does. Use whatever is locally popular as a classified system. Ask local gyms if you can advertise the service with them for people who might want their pets exercised while they work out. Same for restaurants and pubs/bars.

Social networking was a thing before social media and gig proxies. It’s all about finding the people.