This lends me to believe that the pins of the arduino are directly supplying the control signal to the servos.
Is this acceptable? OR do you need drivers?
The motors as specified by Sainsmart are MG966R and SG90
Servo control signals direct from Arduino pins are no problem.
It's only the + and - power to servos that you need to be careful about and they come from the terminal block which is correct. You need to provide a suitable power supply (I imagine you mean MG996R servos and they take a lot of current).
Budget one Ampere per servo and you should be safe. So, a 5-6V, 6 Ampere power supply, or hefty rechargeable batteries would be good choices. Don't forget to connect all the grounds together.
An ATX computer power supply is a cheap source of 5v and 12v power. Remove them from old computers, and jumper two of the appropriate pins to turn them on.