This depends on what issue you have with milk. If it is all dairy then yogurt will most likely affect you the same way. There are several non-dairy milks available on the market...organic soy, rice, hemp, almond, coconut, etc.
If using yogurt in place of liquid, you may need to add a little bit of thin liquid to your Blast too so that the ingredients can blend well.
Yogurt has less lactose than milk so if lactose intolerant you may be able to tolerate, however I'd suggest choosing a non-dairy yogurt instead if this is the concern.