Odds of Getting In
This isn't quite professional baseball, so you don't need to be some one-in-a-million phenom to make the cut. But it's also not like you can just walk into a local animal rescue office and fill out an application. The problem is that it's a relatively new professional field, and there's a lot of interest. A lot of people tune in to Animal Rescue and figure, "Hey, I can save some skylarks."
Unfortunately, wildlife shelters aren't exactly flush with financial resources, so they tend to rely mostly on volunteer labor. There just aren't that many paid positions as professional rescuers, and if you want to fill one of those rare paid spots, it's going to really help to be educated in the field.
A background in ecology and biology is a must, and even then, you should be prepared to start out as a volunteer. A shelter is a heck of a lot more likely to bring you on full-time if you've proven you're reliable and can do the job. Make it through a few months, don't accidentally cage the bobcats with the sparrows, and you'll start to look like a contender for a paycheck.
You can definitely find a job (and get your mitts on that sweet $20,000 salary), but you'll almost certainly first have to build a relationship with a vet, clinic, or shelter. Get your certification, put yourself out there, and be aware that you're going to need experience before you're really marketable.