Assuming that you're semi-legit, you're sure you will do a big volume, and you haven't sent this kind of volume before, I would contact them proactively first. Maybe under the guise of saying something like, "I'm going to be doing a large {launch|new campaign} on $date and I wanted to make sure there will be no problems processing my expected $XX,XXX revenue or handling product fulfillment for this many new customers" or some shit like that.
It's what a professional would do and it gives you a chance to get in on the front foot and feed them some bullshit about where your traffic is coming from etc.