If JASP uses R for doing analyses, is it possible to export the R code which was used for that analysis?

Not without considerable effort. When JASP was first developed, our focus was on producing a GUI that would retain the desirable properties of syntax (such as reproducibility) without requiring or producing syntax itself. We believe we did a decent job on the GUI, but we recognize that some users still like easy access to the R code. Adding this functionality to JASP has been one of our top priorities and we hope to add it to JASP in the near future.