There is only one species of reindeer and it definitely cannot fly! But there are perhaps several hundred thousand species of living organisms yet to be classified, and while most of these are insects and micro-organisms, this does not completely rule out flying reindeer which only Santa has ever seen.
There are 2 billion children (persons under 18) in the world. But since Santa doesn't appear to handle the Shinto, Buddhist, Muslim, Hindu, Jewish, Animist, and atheist children, not to mention the bah-humbug non-Santa believers, that reduces his workload to 15% of the total - about 378 million. At an average census rate of 3.5 children per household, that's 91.8 million homes. One presumes there's at least one good child in each.
Santa has 31 hours of Christmas to work with, thanks to the different time zones and the rotation of the earth, assuming he travels east to west (which seemes logical). This works out to 822.6 visits per second. This is to say that for each household with good children (well, good enough) Santa has 1/1000th of a second to park, hop out of the sleigh, jump down the chimney, fill the stockings, distribute the remaining presents under the tree, eat whatever snacks have been left, get back up the chimney, get back into the sleigh and move on to the next house. Assuming that each of these 91.8 million stops are evenly distributed around the earth (which, of course, we know to be false but for the purposes of our calculations we will accept), we are now talking about .78 miles per household, a total trip of 75.5 million miles, not counting stops to go to the toilet, which most of us must do at least once every 31 hours (In that time he also, of course, consumes almost 100 million pies and about 25 million litres of of sherry - are you surprised you don't see him for the rest of the year!?)
This means that Santa's sleigh is moving at 650 miles per second, 3,000 times the speed of sound. For purposes of comparison, the fastest man-made vehicle on earth, a space probe, moves at a pokey 27.4 miles per second - a conventional reindeer can run, tops, 15 miles per hour.
Now consider the huge load in the sleigh. Assuming that each child gets nothing more than a medium-sized lego set (almost 1 kg or 2 pounds), the sleigh is carrying 321,300 tons, not counting Santa, who is invariably described as overweight. On land, non-flying reindeer can pull no more than 300 pounds. Even granting that "flying reindeer" could pull ten times the normal amount, we cannot do the job with eight, or even nine. We need 214,200 reindeer. This increases the payload - not even counting the weight of the sleigh - to 353,430 tons. Again, for comparison - this is four times the weight of the QEII cruise liner.
353,000 tons travelling at 650 miles per second creates enourmous air resistance - this will heat the reindeer up in the same fashion as a spacecraft re-entering the earth's atmosphere. The lead pair of reindeer will absorb 14.3 quintillion joules of energy. Per second. Each. In short, they will burst into flame almost instantaneously, exposing the reindeer behind them, and create deafening sonic booms in their wake. The entire reindeer team (all 214,000 of them) will be vaporized within 4.26 thousandths of a second. Santa, meanwhile, will be subjected to centrifugal forces 17,500 times greater than gravity. A 250-pound Santa (which seems ludicrously slim) would be pinned to the back of his sleigh by 4,315,015 pounds of force.
In conclusion - If Santa ever DID deliver presents on Christmas Eve, he's dead now.