There are lots of packages for polynomial interpolation in Julia that can be applied to approximate the inverses of expensive functions. See e.g. Approximate inverse of a definite integral - #3 by stevengj
I’m not sure if there is anything like this that is prepackaged to approximate inverse CDFs from data; presumably you want some kind of fit or spline if you have the numerical CDF (via the quantile function in the Statistics standard library, for example) evaluated on a grid. There are lots of packages in Julia for various forms of spline interpolation too, including monotonic Hermite splines — it should be straightforward to apply this to the quantile output, no?
PS. I split this post off into a new thread, rather than resurrecting a 3-year-old thread on a loosely related topic.