Does my 2-month-old need vitamin d?

Yes: Consider vitamin d supplementation (at dose of 400 iu per day) for exclusively breastfed babies.  vitamin d can be given in the form of infant multivitamin drops or vitamin d drops.

Yes & no: There is probably a regional issue involved depending on sunlight exposure in your area.D is activated by sunlight & breast fed babies in areas with fewer sunlight days probably need the 400 iu recommended daily. That is likely overkill in the very sunny parts of the country like my area. Breast milk & sunlight seem to provide what they were designed to do.