Establishing a diverse gut microbiota after birth is essential for preventing illnesses later in life. However, little knowledge exists about the total viral population (virome) present in the gut of infants during the early developmental stage, with RNA viruses being generally overlooked. Therefore, this small pilot longitudinal study investigated the diversity and changes in the enteric RNA virome in healthy infants from South Africa. Faecal samples (n = 12) were collected from four infants at three time points (on average at 8, 13, and 25 weeks), and then sequenced on an Illumina MiSeq platform. The genomic analysis revealed a diverse population of human enteric viruses from the infants’ stools, and changes in the enteric virome composition were observed over time. The Reoviridae family, more specifically the Rotavirus genus, was the most common and could be linked to viral shedding due to the administration of live-attenuated oral vaccines in South Africa, followed by the Picornaviridae family including parechoviruses, echoviruses, coxsackieviruses, enteroviruses, and polioviruses. Polioviruses were also linked to vaccine-related shedding. Astroviridae (astroviruses) and Caliciviridae (noroviruses) were present at low abundance. It is evident that an infant’s gut is colonized by distinct viral populations irrespective of their health state. Further characterization of the human virome (with a larger participant pool) is imperative to provide more conclusive insights into the viral community structure and diversity that has been shown in the current study, despite the smaller sample size.