I am the small animal leader in our club. Not sure if sign up times vary by state or county, but here it is March 1st. Our club starts meetings in August and they stop registrations in January. My kids show or have shown dogs, poultry, rabbits, goats & llamas.
4-H is an excellent program and encourages kids to be leaders, involved in the community, etc. It is not just a country kid thing, their all all sorts of crafts, flowers, veggies, cooking, baking, sewing, shooting sports, quilting, writing, photography, singing, dancing, music, engines, collections, calligraphy, vet science, etc. Then of course, all the animals, cats, dogs, rabbits, guinea pigs, chickens, geese, ducks, turkeys, goats, sheep, pigs, cattle, horses, llamas and more.
There are lots of things for kids to do that live in the city, not just country kids. Most can have guinea pigs, dogs and cats, some rabbits, and chickens in the city. Some programs allow the kids to lease or borrow animals from others. Some of my 4-H kids live in the city and use some of our animals. During the 4-H season they help pay for feed, come out and do chores, learn how to care for them and show them.
Best place to start is your local 4-H Extension Office. Good luck!!