Qualifications
You don't need to be a fully trained vet to work in wildlife rescue, but you also can't just walk in off the street. The position requires training, experience, and certification.
First off, know that most wildlife rescue centers are primarily staffed with volunteers. This isn't to say people aren't getting paid for this work—just that many more people volunteer their time to the animal rescue cause (source).
If you're hoping to do this work and get paid for it, you're probably going to need a college degree in biology or ecology to really compete (source). You can get licensed without a college degree, but having that diploma in a relevant field is a major leg up in this field.
There are also schools that offer classes in wildlife management, behavior, ecology, and field techniques. More than sixty colleges in North America offer accredited veterinary tech or animal health tech programs (source). In other words, you have some options.
Whether you go the college route or not, though—and whether you're hoping to pay your rent by rescuing animals or simply spend some quality time volunteering—certification is generally expected. Different states have different requirements, but the Certified Wildlife Rehabilitator certification exam is a commonly known and respected certification in this line of work (source).
Once you pass, you'll need to re-certify every two years, as well as prove that you have "continued your education" by either attending wildlife conferences or trainings, presenting at conferences, or publishing papers in wildlife-related journals.