Aristocratic women would, most of the time, employ wet nurses, though they normally brought them into their households rather than sending their babies out to the wet nurses' house. How widespread wet-nursing was among urban elite women is less certain - abundant evidence exists for the practice in late medieval Florence, yet virtually none for late medieval London. It seems like some wealthy Jewish families would have employed them, since there was a decree at the Third Lateran Council in 1179 forbidding Christian wet-nurses from nursing Jewish babies, on the pretence that Jewish parents forced wet nurses to "express" their milk after Easter lest the baby ingest the Eucharist. As for lower down the scale, peasant women and the wives of artisans, shopkeepers and labourers would have undoubtedly breastfed their children i.e. the Anglo-Saxon treatise Lacnunga prescribes a special charm for a mother who cannot lactate. And in the abstract, most authors of medieval scientific treatises seem to have assumed that it was a natural purpose of a human mother to breastfeed her child i.e. Bartholomew the Englishman wrote in the 13th century that "the mother [mater] is so called because she nourishes the newborn with her breast [mamma]."

I'd highly recommend reading Henrietta Leyser's "Medieval Women: A Social History of Women in England, 450 - 1500" (1995) as its got an excellent chapter on motherhood in the later middle ages (1100 - 1500).