Anything can be fitted to anything else, if your engineering skills are good enough. But unless you can do welding I don't recommend that you even try - it's not like it will bolt straight on. On top of that there's the issue with the drive train. Mopeds and scooters all have gear boxes to adapt what the engine puts out to something that the vehicle can use. Most lawn mower engines won't have that.
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Well first off, a lawn mower is probably not the best choice. The cheapest way to attach it to the axle is to turn it on its side. (the jack shaft is usually vertical.)This is not very effective though. It is likely to cause piston seizure, and gas injection fail. Try going to a scrapyard and finding a ATV. You can salvage the engine, wheels, shocks, and sprockets off of it. The engine will probably range from 20hp to 50hp. If you think that is to big of an engine, Try going to a small engine repair shop. You might be able to get a used 5-10hp engine for cheeper than normal.