Like everything in the market world it's supply and demand: jobs that are easy to do don't pay, jobs that pay well require expertise and financial outlay. But it's not impossible, people do do it. I just wouldn't expect to make much money at first, and lots of people even with experience and space find they can only just cover their costs.
The easiest fish to breed are probably guppies, platies, swordtails, convicts, kribs. Then maybe bristlenose plecs, danios, cherry barbs, corydoras.
Most lfs are overrun with livebearer fry, convict and krib fry; these are the rabbits of the fishy world, and anyone who keeps them is usually desperate to get rid of the fry. And many local shops also breed their own, as it requires virtually no effort. So the store owners are unlikely to pay you, unless you can establish a good relationship with the shop. Or alternatively, go into breeding special colours or shapes. IF you do want to breed on a regular scale, you will need plenty of space for grow out tanks. Livebearers eat their fry, kribs when spawning beat up any other fish in the tank.
My local shops also seem to have a regular supply of baby bristlenoses. It might just be possible to get paid for these if you had the space to rear them to a sexable age.
I would start with something easy, to get the experience. Maybe a livebearer first and then some of the easier egglaying fish. I would imagine (no personal experience) the money would be in breeding something like catfish (maybe some of the smaller L numbers) or other fish that are not so often bred, but that there is still a regular demand for.