Can multiple Arduinos be connected to a single relay?
Looking at the schematic, here are some points to be kept in mind when using this relay board:


The relay board should be powered through the 12V connector
The relay board has its own 5V regulator
The optocouplers are attached in "reverse"


So, the external connections should be


Connect all the arduino grounds together, and connect them to the board ground
Do not connect any +5V wire together
Connect arduino 1 to IN1, arduino 2 to IN2 and so on
Writing a HIGH value on the INx pin will result in the relay to be de-energized (so contact goes in NC position), writing a LOW value on INx pin energizes the relay (so contact goes in NO position)


If making this connection the board does not work as expected, try to detach all the arduino boards and manually shorting IN1 and +5V, IN1 and GND, IN2 and +5V, ...