Solid State Relay operation
It will draw about the same current at 5V. The IR LED used in the SSR requires a certain amount of current to switch reliably and the typical input circuit for a 3-32V input SSR is a simple constant-current regulator made with discrete BJTs. 

Here is another manufacturer's graph showing input current vs. voltage: 



Of course this is just an educated guess on my part as the datasheet is characteristically inscrutable and/or mute on the point (as well as lacking safety agency approval markings), but every similar SSR I've seen works similarly to this.