All cartridges have recoil. Even .22 Short.
Bullseye(ISSF is the authority now) is shot with .22's. Both rifles and handgun. Serious target firearms can get expensive, but they don't have to be. Any semi-auto .22 pistol with adjustable sights will do. Rifles usually have heavy barrels and target sights on 'em. A Savage Mark II FVT is an example of a lower priced(MSRP is $479US) target rifle.
Ditto for Canadian Shooting Sports Association and Shooting Federation of Canada bullseye matches.
Where you are matters as these matches are all club level things. Great fun too. And you don't have to be good to have fun.
There's also Biathlon. Skiing and shooting in winter. Cycling or running and shooting in summer. Biathlon rifles are usually mag fed, straight pull, bolt action, .22 target rifles. Any .22 target rifle will do though. A purpose built biathlon rifle starts at a bit below 2 grand.