Ok, so it' s not a rifle, but it really thinks it is, Marlin leveraction .410, use shotshells for rabbits and slugs for coyotes, it' d be about perfect for what you' re asking, hitting rabbits on the run with a rifle is a difficult task, shotshells are the answer. They also work great on coyotes, but most folks don' t like them in as close as I do, so slugs are a better option for most.

Otherwise, maybe try on a .22mag like in a Ruger 10/22mag. Any .22centerfire bolt gun would suit you well though, it just depends on what the gun will be, a bunny gun that does coyotes, or a coyote gun that does rabbits...if it' s primarily going to hunt bunnies, then get the rimfire, if primarily coyotes,then get the centerfire...the .410 would suit you great either way.