Controlling pull-up pins from external circuit
This is most easily done if two conditions exist:

1) Your Arduino board runs from 5V (instead of 3.3V)

2) You have extra output pins on the Arduino board that aren't being used.

If both of these conditions exist, simply connect the grounds of both boards to the same point and then connect each input pin of the FX board to the Arduino.  These output pins are set HI when not active (during initialisation). 

The only difference is that now you will activate 2 pins when you want to do one of your effects: set the pin driving either the door lock or strobe HI, then set the pin connected to the FX board LO.  Return the pins back to the non-active state when finished.

There is a possible advantage to doing things this way: each of the outputs can have different active time.  In other words, let's say that the FX board will repeat the sound over and over again so long as the input is held active.  Maybe it does, maybe it doesn't.  But let's assume that it operates this way.  

But: you need the strobe to be active for several minutes - far longer than the sound.  

Because the outputs are separated, you can set the pulse duration for each output independently.



[Edit]

Based on your comment, I see that you want to have the same pin on the Arduino control both your effect (door lock, strobe) as well as the sound effect.

That's also easy but requires one extra transistor per FX input.  The reason for the extra transistors is that your relay card is active HI but your FX card is active LO.  The transistor provides logic inversion.

You can use either bipolar NPN transistors or N-channel MOSFETs.  The MOSFETs are nice in that they don't require base resistors but they may cost a few pennies more than a bog-standard NPN transistor.

If using NPN transistors such as 2N4401 or 2N2222, connect Emitter to ground, connect Collector to the FX input pin, connect the Arduino output pin to both your relay card and to one side of a 10k resistor.  The other side of the 10k resistor connects to the transistor Base.

If using a N-channel MOSFET such as the 2N7000, connect Source to ground, Drain to the FX input pin, Gate to the same Arduino pin that is driving the relay card.

In either case, when the Arduino pin goes HI to turn on the relay, the corresponding transistor turns ON and pulls the FX input pin LO to trigger the sound effect.

Easy as pie!