It would be a variation on "Did they purchase it on the open market on my behalf, or just allocate some of their own units to me?"
First up, it appears that you do not understand who "they" is. Your broker handles the dividend reinvestment, not the ETF manager or authorized participant. And your broker doesn't begin to have the authority to "dilute the stock by just increasing the amount of trade-able units".
What I certainly would expect to happen is that because all investors in that ETF get paid the dividend at the same time, your broker pooled together the money from all the people who chose to reinvest, bought a large number of units, and divvied these up (which creates the possibility for DRIP to result in fractional shares)
If DRIP purchases affect the supply-demand curve sufficiently to affect price, then ETF arbitrage will kick in -- the authorized participant will buy up the underlying stocks, form a basket, and trade it to the fund manager in exchange for additional units of the ETF. The authorized participant will then sell these to your broker at just barely above "net asset value". Since the holdings of the ETF increased, there's no dilution going on.