You could always use a generic quadrature routine, though that will have suboptimal efficiency if you have many frequencies x_k at which you want to compute F(x_k).
Technically, that paper describes a Bluestein’s (1968) algorithm for the “Zoom FFT”, generalized to the chirp-z transform by Rabiner and Schafer (1969), which is not the same as the continuous Fourier transform (note: Bailey and Swarztrauber’s terminology was not widely adopted, because a fractional Fourier transform already meant something else). (It computes a sum, not an integral.)
You can use this sum to approximate a continuous Fourier transform of a rapidly decaying function if you sample finely enough and normalize correctly, of course.
I don’t know of a Julia package for this offhand, but it’s pretty easily to implement given an fft function (e.g. from the FFTW.jl package), since it’s just a convolution that can be written in a few lines of code.
More generally, if your frequencies are not equally spaced, you could use a nonuniform FFT, e.g. FINUFFT.jl or FastTransforms.jl.