I used inkjet machines with a remote printhead in previous jobs , printing on paper, cardboard and plastic food packaging .
The technology I used was called CIJ (continuous ink jet) , the ink and a solvent was mixed in the machine and pumped to the remote printhead ( 2-10 metres away) ,where a piezo nozzle turned the ink stream into droplets which were then electrically charged.
An electrically charged deflector plate then directed the ink droplets into text and onto the print substrate .
Any unused ink drops were deflected into a "gutter" and returned to the control unit by vaccum to be reused.
The machines were designed for continuous industrial use and even basic units were expensive to buy , €7k approx .
Linx and Videojet were the machines I used.
I have seen handheld inkjet printers used on cardboard and similar packaging and had a demo of a Reiner handheld inkjet printer which may be more suited to your application but still quite expensive.