Enables the directed movement of nicotinamide mononucleotide into, out of or within a cell, or between cells. Nicotinamide mononucleotide is a ribonucleotide in which the nitrogenous base, nicotinamide, is in beta-n-glycosidic linkage with the c-1 position of d-ribose. It is a constituent of NAD and NADP.