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

Yes, this is possible. When JASP was first developed, the focus was on designing a GUI that would retain the desirable properties of syntax (such as reproducibility) without requiring or producing syntax itself. Since then we came to the realization that many users would like to have access to the R syntax, and it is available in JASP from version 0.17 onward. The R syntax is visible by clicking the analysis-specific R icon.
At the moment, the R syntax works only within JASP itself — it can be used to control the GUI and it runs in the JASP R console. In the near future the R syntax produced by JASP will also work in R Studio.