From memory, have a look for RCurl - that enables R to issue http calls and grab the data returned.
With that, R should be able to call JIRA's REST APIs to pull data from JIRA (you'll need to find a JSON library for R as well, that's where my memory fails me)
Custom fields are returned when you ask for issue data.
Have a look at https://developer.atlassian.com/jiradev/jira-apis/jira-rest-apis/jira-rest-api-tutorials/jira-rest-api-example-query-issues and the documents alongside it.
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