Mowers did come with chain driven axles but they usually have a differential built in.
Have another look at the axle and see if it has a big round part near the sprocket.
If it's working properly the wheels will spin in opposite directions if the machine is off the ground.

AMF and MTD made small machines like this, maybe others.
They usually have a vertical shaft engine which drives a belt to the transmission, then a horizontally oriented output shaft driving a chain to the rear axle, which has a differential.

They aren't really very strong. Sometimes they don't even have axle bearings, only bushings.
I wouldn't recommend making it spin faster or giving it too much power. If you want a racing mower build real solid axle with bearings. Basically a go kart with a tractor body.

 

My old Bready has a horz shaft engine belt driven to a cross mounted normal trans, then chain drive to a live axle. [it does have a dif built into the axle though....] so they were made, but unless its something real small riding mower, I think mostly a long time ago. [mine is a 1960]

 

What do you mean by "live axle". I've seen lawn tractors with solid front axles and pivoting rear axle (which makes a lot of sense).

 

You might consider how a motorcycle rear axle pivots upon an A frame, and a spring & shock is used to hold it all up. I suppose a guy could mount the transaxle or whatever & possibly the engine also, to the 'A' frame, or various drives, chain-belt-shaft might be used, depends how exotic you want to get. The basic 'A' frame at least you would give the UP & down flex, not so much side to side tho. You need a differential for those tight turns and power-sliding. If your front axle pivots [side-to side]that helps a lot to control steering, and is practically necessary.

I got to ride a go cart about 40 yrs ago that was set-up just that way, spring loaded pivot in the middle of chassis and engine & chain driven differential floated as an solid assy. in the rear, was great for rough pastures and little jumps.

wwxx