Yes, a baby's الحج is valid and the parent earns reward — use a carrier instead of a stroller, plan for cooler hours, and bring all supplies.
Yes, performing الحج with a baby is جائز. A woman brought her child to النبي (صلى الله عليه وسلم) and asked: 'Is there الحج for this one?' He replied: 'Yes, and you will have the reward' (Muslim). The الحج performed on behalf of a child is valid and earns reward for the parent, though the child will still need to perform their own واجب الحج after reaching puberty.
Practical considerations are significant: the extreme الحرارة and massive crowds during الحج present real challenges with an infant. Use a front-facing baby carrier rather than a stroller for الطواف and السعي, as strollers are nearly impossible to navigate in peak crowds. Bring ample supplies — diapers, formula/food, الأدويةs, sun protection, and a lightweight blanket. Plan your rituals during the early morning or late evening when temperatures are lower and crowds are thinner. Keep the baby hydrated frequently. Diaper changes and breastfeeding breaks may require stepping away from the ritual temporarily — this is completely جائز. Ensure your الحج group is aware you have an infant so they can provide appropriate support.
Source: صحيح مسلم, Kitab al-الحج; ابن قدامة, al-Mughni