Wiring 100 Peltier Plates Together
The simple answer is: maybe. There are a few things you've skipped over. First is the plate thermally insulated? If the plate is covered in thermal insulation, you can certainly cool it to 10 C, and probably get down to about -20 C if the insulation is very good.

100 30-watt TECs will draw 3 kW. A general rule of thumb with Peltiers is that you can get 10% of that in cooling. So. Let's run things backwards. 10 C is about 15 C below room temperature. If you were to run 300 watts of heating into your plate, could you get a 15 C rise in temperature? If so, you have a pretty good shot at cooling it to your specifications. On the other hand, if it's exposed to air, it may be hard to get the effect you need.

It's not reasonable to think that you can pack your TECs perfectly. As KalleMP has pointed out, you need to leave some room to run wires, but let's say you used a 9x9 array of coolers, with about 5 mm between units. Then you would have 81 coolers, with a power dissipation of about 2400 watts, and about 240 watts cooling. Let's assume your controlled plate is perfectly insulated. Let's also assume your plate is 6 mm thick, rather than 3, so as to let the metal spread the temperature evenly. Then the volume of the plate is .015 cu meters, and the weight is 40.5 kg. The specific heat of aluminum is .9 J/deg-kg. To drop the temperature of the plate 15 C will take an energy of $$E = 0.9 \times 15 \times 40.5 = 547 \text{J} $$ and at 240 watts it will only take about 2.3 seconds to drop the temperature you need. The less the insulation, the more power you'll need, and keep in mind that Peltiers do not produce much temperature differential when run at high current.

Your construction approach is appropriate, but as has been mentioned, in principle you might need a pretty good cooling system to handle 2.4 to 3 kW of waste heat.