Here's how the ssr board input is wired, just so others don't have to go find the pdf.
But you are correct about needing a pullup resistor for each output of that I/O expander.
It can only pull high with ~100uA which isn't enough to switch on the NPN in the relay board. A 10K resistor to VCC would fix that.
Or you'd could find a I/O expander that can source current, then you waste very little energy and dont need the pullups.