How to calculate required voltage + amperage for a linear servo array?
Using a voltage regulator to drop to 3.7 V is a bad idea, that would generate (12V - 3.7V) * 4.5 A = 37 W of heat.  However, note that 3.7 V is the voltage of a Li-ion cell, which is great for you, because Li-ion batteries are the standard for energy-density per unit weight in batteries.

The capacity in mAh will be determined by how long do you need your battery life to be.  Unless your actuators are stalled all the time (bad), you will only need 150 mA / actuator while they're running.  150 mA * 10 = 1500 mA, which means that if you run all 10 actuators continuously for 1 h, you will use 1500 mAh.

I would probably try a single modern 18650 cell, they can provide up 3.7 V at up to 20 A: with a capacity of 3000 mAh, a single one of these batteries would be (nominally) able to run all ten actuators continuously for 2h.

Be careful, though:


Li-ion cells should have protection circuitry to prevent under-voltage or over-charging.
I have not built anything with 18650 cells before.  I'm suggesting them because they have the right rating for your task, and they're really popular (probably the most popular type of Li-ion cell).
Not all 18650 cells are created equal.