It is permissible for a woman to breastfeed her child before doing ghusl
Question

                If a woman has not yet done ghusul can she breast feed her child?
Praise be to Allah. 
          It is permissible for a women to breastfeed her child 
          before doing ghusl, whether it is ghusl after nifaas, menstruation or 
          sexual relations. There is no evidence to indicate that she must do 
          ghusl in these cases in order to breastfeed her child. Rather ghusl 
          is obligatory in cases of impurity in order to pray or to do any act 
          of worship for which tahaarah (purity) is required. The woman who is 
          bleeding following childbirth or menstruating must do ghusl when the 
          nifaas or period ends in order to do those acts of worship and also 
          so that it may become permissible for her husband to have intercourse 
          with her. It is not haraam for the woman who is menstruating or bleeding 
          following childbirth to touch anything during her period or nifaas, 
          except for the Mushaf (copy of the Qur’aan in Arabic only). And Allaah 
          knows best.