Does dividend reinvesting influence the price?

When brokerages auto-reinvest these proceeds (DRIP), won't that influence the price (assuming that it's a market order)?

Yes and no.

Yes, in theory. If the mass of buy orders is big enough (i.e. enough dividends are reinvested like that).

No, because they may have stocked up awaiting the gain.

For reinvestment of dividends, brokers combine the expected cash distributions to be received from dividends and they then purchase shares at the open on the payable date. Some studies indicate that this creates buying pressure that results in an increased share price as the dividend payout date approaches. Some authors suggest that share price then drops after the payable date.

OTOH, traditional Dividend Reinvestment Plans (DRIP) with the company, the dividends purchase the shares directly from the company's treasury stock. When sold, such shares are sold back to the company. So these DRIPs have no effect on share price.