The phrase “on tick” means: On credit.
Years ago, when this commercial term first came into use, it indicated a written document, a form of IOU.
That is, it was just a contraction of “on ticket,” and the “ticket” was some form of note of hand, or acknowledgment of indebtedness.
The contracted form came into use in the fore part of the seventeenth century.
The Oxford English Dictionary cites a usage of 1642: “They would haue . . run on tick with Piggin for inke and songs, rather than haue lost the show of your presence.”