Can you imprint a card onto Chrome Mox and then use its mana to pay for its own imprint cost?

Chrome Mox's imprint is a triggered ability that triggers when Chrome Mox enters the battlefield. At that point, Chrome Mox is already on the battlefield and its mana ability is already available — but the imprint trigger goes on the stack and you must pay nothing to put a card into exile from your hand. There is no 'cost' to imprint; it is a free triggered ability that asks you to make a choice (exile a nonland, non-token card from your hand).

Because imprint has no mana cost to pay, the question of using Chrome Mox's mana to pay for it is moot — there is simply nothing to pay. Triggered abilities do not have casting costs (CR 603.1, 112.3b). You activate Chrome Mox's mana ability by tapping it (CR 605.1a), and separately, the imprint trigger resolves asking you to exile a card.

The deeper timing issue: Chrome Mox's mana ability cannot even be activated until it is on the battlefield (CR 605.3b). Its imprint trigger is placed on the stack when it enters, and the game does not pause in the middle of that trigger's resolution to let you activate mana abilities to 'pay' for it, because again, no payment is required.

Concrete example: You cast Chrome Mox and it resolves, entering the battlefield. Its imprint trigger goes on the stack. When that trigger resolves, you simply choose to exile a nonland, non-token card from your hand (or choose not to). You could tap the Mox for mana in response to the trigger sitting on the stack, but that mana would go into your mana pool with no imprint cost to spend it on.

In summary: imprinting a card has no mana cost, so there is nothing for Chrome Mox's mana ability to pay for. The two abilities are independent, and no 'bootstrapping' scenario applies here (CR 603.1, 605.1a, 112.3b).