When you say wire transfers, do you mean international wire transfers or domestic ACH transactions? For the latter, there are a few companies that provide electronic ACH processing & an API.

The biggest problem that you'll face is not actually moving the money, but complying to anti-money laundering regulations and the like.

That said, we're building a banking service with an API. We've done the hard work. I'd love to know more about your use case to see if we can help.