Officially, no: there is no API for File Manager. It's a relatively new feature, and staff mentioned something to the effect of that they were going to wait and see how it gets used "in the wild" (and how app/driver access might work--since it's a shared space) before considering adding features like this. I haven't seen any mentions of anything beyond that at this point.
However, unofficially, some people have "reverse engineered" a way into it, I assume by basically doing the same HTTP commands you'd be initiating if you were using a browser in the admin UI yourself. Here's one:
I also know Hubitat Package Manager did something similar for file access (though that is by no means the primary goal of this app):
But again, there is nothing official, so the above may not work 100% of the time or may break in the future. But it's possible something native will come (or not).