Is this normal? Would a duino output pin have enough punch to power this thing on just 5V? Should I use the alternate power source instead? If so, again, does anyone know which voltage and current are required?
You could connect the relay module's VCC terminal to Arduino's 5V to power the module.
From this link it shows coil power: 0.36W, 0.45W.
Using P = V2/R, 0.45 = 25/R, R = 55 OHM (coil resistance).
Using V = IR, I = 5/55 = 0.091 (91 mA is the coil's current draw).
So both relays energized should draw less than 0.2A from the Arduino's 5V supply.
You should be OK ... it would be the same as having about 9 LEDs on at 20 mA each.