Sometimes, but not always. Again, this varies widely depending on your local league's policies, so check with them. However, most soccer matches will not be postponed due to rain, especially if it starts raining while matches are going on. The only times you usually see matches postponed or stopped is due to severe weather (thunder/lightning, strong winds, etc.) or if it has been raining so much that the fields are unsafe (cleats don't grab anymore). Many leagues will postpone matches if it has been raining for some time before match day because the fields have standing water on them or the ground is so soft, it will damage the fields to play on them. Each league has their own 'threshold' and many are beholden to the decision of municipal parks departments who will make the decisions.
But more often than not, if your child is playing and it starts to rain without thunder/lightning, the match will go on. So if you know rain is forecast for a match day, bring dry clothes, towels, and dry shoes - your child will be glad you did!