Using curl, it would look like this:
curl -k -u admin: -X POST -H "Content-Type: application/json" https://localhost:17778/SolarWinds/InformationService/v3/Invoke/Json/Cirrus.ConfigArchive/Download -d @download.json
Where download.json contains something like this:
[ ["ef324766-a202-4438-81bf-2c8cbf527743"], "Running", "ignored" ]
The GUID that appears on line 2 should be the value of Cirrus.Nodes.NodeID for the nodes you want to download configs for. You can have more than one of these. The second parameter (on line 3) is the config type. The third parameter (on line 4) is ignored.
[Update 6/20: Corrected URL]