It is possible to start right out of the chute. However, keep in mind that as an agency nurse, the facilities that you will go to will expect you to have an existing knowledge base: you're not going to get the full orientation that one of the facility's new grad (or just plain old new) hires may get. It's going to be mostly an orientation to the units and procedures, and then they expect you to run. And you need to have an idea of what you're doing as a nurse, since it doesn't always take much for you to be tagged a "do not return" at a facility.
Also, agencies may provide basic information and some training, but they're not going to rehash nursing school for you. If you've never worked L&D before, the agency isn't going to offer you L&D 101; they expect you to already know about L&D. Hence why a lot of agencies prefer nurses with some experience.
Like the prior poster stated, you have nothing to lose by asking! And perhaps there is an agency out there that would offer you L&D 101 or otherwise provide nursing training but they're not too common. So keep all this in mind should an agency decide to take you on.