This is a continuation of a thread started in the old forum - here. I thought I'd tie up some possible solution and which give a nod to importing JSON in general by moving it to CSV, which is supported - see more on CSV/tab-delim import to Tinderbox.
As at v6.6.5, I believe there is no import parser for JSON (though there is on the export side - with jsonEncode()).
Trello exports JSON - see here. That article mentions a Chrome (only) plugin that lets basic level Trello export CSV, which Tinderbox can handle; Trello 'Business class' level account can export CSV directly. It is worth noting that the Trello article says the JSON export is quite complex and likely more data (e.g. internal Trello relationships) than you'll likely need in the likes of Tinderbox. The latter seems a reason engineering in specific Trello support.
Tech folks happy with the command line could install a script (get
json2csv from github here). Then, save to JSON (from Trello or other apps), pass through
json2csv and drag the CSV into Tinderbox to invoke its CSV import (see link above). You could also knock up a shell script to automate that and even invoke it from within Tinderbox by using a runCommand() call in a stamp.
Less tech folk can use an online JSON -> CSV utility, such as here. Then the flow is, save JSON file, upload to website, covert, copy CSV output to the clipboard, switch focus to Tinderbox and paste the (CSV) data into their TBX document.
I forgot (hat tip for reminder: @PaulWalters) that less tech users should be aware that if you use a web service you are using someone else's server so it might not be sensible if you have sensitive personal data in your JSON.