2-Channel Relay Question
The control circuit on the relay board is poorly designed.  

  

If you apply 3.3 V then there's 2.6 V across R1, so the current will be 260 &micro;A. Across R7 we have 0.7 V, so that's 70 &micro;A through R7, leaving 190 &micro;A for Q1's base. If Q1 has an \$h_{FE}\$ of 100 (typical for a general purpose transistor), then you have 20 mA collector current. The schematic doesn't give a value for R2, but even at a few mA through the LED there's not enough current left for the relay.  

The solution: replace R1 with a 1 k&Omega; type, which will give you 2.6 mA base current, and 260 mA collector current. That should be enough to drive both LED and relay.

The reason it works with 5 V is that at 5 V you have a higher base current, a bit the same effect as decreasing the value for R1. Apparently the design was not tested at 3.3 V.

BTW, remove R7, it serves no function. It only takes current away from Q1, which you don't want.