True, R can read SPSS files, however, you need to install the appropriate packages before you can run your analyses. The help feature can very useful. But you need to be more specific about what you would like to do.
Maybe it is more interesting to stand with just one, for organization porpoise. SPSS, JASP and R are consolidated tools, and an ANOVA should be no problem to be done in any of them.
I wonder what does "in conjunction" mean in this case? Of course you are able to read SPSS .sav files into R and R can save in nearly all file formats with according packages. JASP can read SPSS files and all other text files and tables.
But with "in conjunction" maybe you mean, R scripts in SPSS or JASP or vice versa? With some preparations you are able to run R scripts in SPSS (see link by Daniel) and you are able to use/install some R packages to use them via the SPSS GUI. With the latest version of JASP you are able to run R code and scripts, but I haven't figured out, how to use packages (maybe someone knows? I didn't spent much time on it and it is in the beta version). I am not aware of any possibility to run SPSS syntax in R (and why should you....).