Can I power two linear DC regulators, each running a motor, from a single 12v source?
Here's the circuit of your regulator board (found here):-



This is a poor design. Input bypass capacitor C1 is the minimum size required to maintain regulator stability. A large electrolytic capacitor should be connected across the power supply input to handle surge currents, particularly if using long wires.

The recommended value for feedback resistor R2 is 240&ohm;. A slightly larger value will still work, but 1K is pushing it. This may explain why the specifications say minimum current draw is 5mA (since normally the 240&ohm; resistor does this). 

They also say maximum current draw is 1.5A, but that is a lie. At 1.5A the LM317 drops 2.25V minimum, so the lowest possible power dissipation is 2.25V * 1.5A = 3.4W. When regulating a higher voltage at this current it would get burning hot in seconds and shut down.

Your motors may only draw 250mA when running, but the starting current could be 5 times higher. If your power supply is able to handle this the voltage may still sag as the motor starts up. This is another reason for using a large filter capacitor across the regulator input.

With two motor going at once you may get interaction between them as one effects the power supply of the other. If this is a problem then you could  try isolating them with a diode in each positive supply lead, like this:-







simulate this circuit &ndash; Schematic created using CircuitLab

The purpose of the diodes is to prevent voltage sag caused by one motor from discharging the capacitor on the other motor.