Hi, JASPAR now provide a REST API to get data from the latest release (2018) and also older releases of JASPAR. Here is the API http://jaspar.genereg.net/api/ and here is an interactive documentation http://jaspar.genereg.net/api/v1/docs
You can also access JASPAR 2018 database by using R/Bioconductor package https http://bioconductor.org/packages/JASPAR2018/
P.S. Although, it is a late answer, but I'm posting this if someone else have similar questions.
You can programatically access the JASPAR 2016 database using this R package: https://bioconductor.org/packages/release/data/experiment/html/JASPAR2016.html