Do I need the Privateer to sell fuel, ammo, weapons, and gear, or can I do that with my normal Kraken (space permitting of course)?

The regular Kraken can only refuel, rearm, and resupply ships landed on its pads or in its hangars. These supplies come from the Kraken's own stock or cargo grid, so in a very rudimentary sense it can, but it doesn't have any means of safely charging customers.