How would I go about getting this Zefal pump to work with a Schrader valve?
It's quite likely the head is dual type switchable. 


Unscrew the retaining cap.
Remove the rubber thing. You might need some gentle encouragement with pliers. 
Spin it 180. If the other hole is Schrader sized, You're in luck.
There will be a plastic widget under the rubber bit. This also needs to be flipped 180 to expose a pin. 
Put the head back together. Widget, pin up > Rubber Thing, big hole up > Screw Cap.






Not something you want to do if you're working with varied valve types all the time, but as you say your family has one type.