When I ordered extra GRE scores online on 1/9, the actual 'report date' on the GRE website wasn't until 1/11, and my two schools had it marked as received by the 1/15 deadline. I think the GRE system takes a few days to process the requests before sending them out (but I think it's electronic, so it should be same-day.) I think that a lot of these schools have leeway around parts of your application you don't have control over, like letters of recommendation or GRE scores, so I wouldn't worry too much.