Arduino.cc forum topic 229646, “Arduino Sensor Shield v5 (APC220) manual”, is a thread about lack of documentation for that shield. However, post #10 in that thread states that pulling off the SEL jumper allows you to use an external power supply to power the devices you attach to the shield. Also see Sensor Shield (Version 5X) Separate 5V for Servos at yourduino.com.
The Robotics Stackexchange question Wiring & driving TowerPro SG90 servos agrees with the 750–1000 mA current draw of SG90 servos. You could go with a 5 or 6 V power supply, with say 12 A rating if you expect to drive 12 servos at the same time. However, as a 12 A supply might be expensive, you probably should attach two or three servos to the shield and measure the current draw. If, for example, your application rarely draws over half an amp per servo, you might scale back to a 10 A or 8 A supply.
Note that an Arduino Uno or Nano (in general, any based on ATmega328-like MCUs) has six pins that support hardware PWM, so you may need to use a Mega instead, or two Uno or Nano Arduinos. (Software-generated PWM signals are another possibility but I doubt it would work well enough if your app is at all complicated.)
Edit 1: If the system is battery operated, you probably should use a 2S LiPo RC battery, that is, a battery with the voltage of two lithium polymer cells in series (so, 7.4 volts) and whatever number of cells you wish to pay for in parallel, as often used in radio controlled devices.
Some LiPo batteries support 25C discharge rates, ie, can discharge at 25 times their one-hour-discharge rating. Thus, a 500 mAh battery might work fine – since 25 times .5 A is 12.5 A – if you only need to run your system for up to 2.4 minutes between battery charges. Or, for example, a $13.49 Turnigy 1000mAh 2S 20C Lipo Pack HobbyKing RC Battery would work ok, even though only rated for 20C. It could power a 12 A draw for 1/12 hour, or 5 minutes.
For the SEL jumper, look in the lower left corner of one of the pictures from your amazon.com link, as below. With the jumper (the short-metal-strap-and-black-plastic-shell assembly, next to the letters SEL) removed, connect battery+ and battery- to the screw-down terminals at bottom left in the photo. Battery+ goes to Vcc, battery- to Gnd.
Regarding a second Arduino, you could split the processing across two of them so that each one only needs to know the current time; or could communicate between them by an SPI, I2C, or RS232 interface; or could move up to a Mega2560 Arduino, which has a lot more PWM channels.
Edit 2: If battery weight is an issue, you may be able to use lower-capacity (hence lighter-weight) batteries if your software only moves a few of the servos at the same time. For example, if you move three or four at a time, and if most of the servos are idle most of the time, you might be able to use a light-weight cellphone battery.
Addicore.com has some useful current measurements for the SG90:
Voltage 4.0V to 7.2V, 4.6V - 5.2V nominal
Running current with 5V supply (no mechanical load) 220 ±50mA
Stall current with 5V supply (horn locked) 650 ±80mA
Idle current with 5V supply 6 ±10mA
Weight 0.32oz (9g)
However, you probably need to make current measurements on your own to see how big the battery needs to be.