For the HCP 500 Release, unprocessed and processed data for a subject can be accessed via REST calls apart from using other modes such as Connectome In a Box, Aspera, Amazon AWS.
In order to access data for a subject, you will need to register at https://db.humanconnectome.org
To get data via REST, in a script, you could use tools like curl.
Details about file and folder organization are available here.
To download a particular file, say the minimally processed diffusion file, T1w_acpc_dc_restore_1.25.nii.gz, the for a subject, say 100307, you could invoke curl as:
curl -u DB_LOGIN_ID_HERE:DB_PASSWORD_HERE -X GET