Does LibreOffice Calc support JSON file importing/sorting?
Had the same problem myself (that's how I found this question).
So, for the next person finding this - the answer is no - LibreOffice Calc does not support direct import of JSON.
And the next logical step indeed is converting to CSV. There are free online JSON to CSV converters, and using one of them (http://www.convertcsv.com/json-to-csv.htm), I was easily able to make a correct CSV which Calc imports without a problem.
One possible caveat is if you have complex objects represented in JSON - that may not be convertible to CSV, but then again, if it doesn't fit into CSV, it probably doesn't fit into spreadsheet format either.
There's a LibreOffice GetRest plugin with documentation written in broken English, that has a "parseJSON" formula. It won't convert JSON to CSV (without a lot of grunt work) but it might help your use case.
If you can run Python scripts in Libreoffice Calc, then it should be possible when you see what's in here: http://blog.appliedinformaticsinc.com/how-to-parse-and-convert-json-to-csv-using-python/