Catalysis of the reaction: N-acetyllactosamine + 3'-phosphoadenosine 5'-phosphosulfate = 3-sulfo-N-acetyllactosamine + adenosine 3',5'-bisphosphate. N-acetyllactosamine residues are found in a number of different carbohydrate types. N-acetyllactosamine can also be written as Gal-beta-(1,4)-GlcNAc.