Looks like the spec presented in relay board is a bit misleading. They only show the need for 15-20mA to drive the relay. I agree if you look at the datasheet for the relay in the picture, 70mA minimum is needed.
There is a maximum wattage the regulator can deal with, which is roughly calculated as (Vin - 5V) * current load (in amps). The higher the input voltage the less current you can draw. Most current available from the regulator is when powered by 7V.
You have several other loads on the regulator besides just the relay board. Does anything else shutdown or just the relays?