It seems that there is not much research concerning the anatomical localization of primitive reflexes. For instance, Schott & Rossor (2003) state in the conclusion of their review

The complexity of many of these responses makes it perhaps unsurprising that detailed anatomical localisation, despite the availability of structural and functional imaging, has in general not been possible.

However, at least for the grasp reflex, they discuss an involvement of frontal areas.

References
Schott, JM Rossor M (2003). The grasp and other primitive reflexes. J Neurol Neurosurg Psychiatry 74(5): 558–560. doi: 10.1136/jnnp.74.5.558 PDF

It seems that there is not much research concerning the anatomical localization of primitive reflexes. For instance, Schott & Rossor (2003) state in the conclusion of their review

The complexity of many of these responses makes it perhaps unsurprising that detailed anatomical localisation, despite the availability of structural and functional imaging, has in general not been possible.

However, at least for the grasp reflex, they discuss an involvement of frontal areas.

References
Schott, JM Rossor M (2003). The grasp and other primitive reflexes. J Neurol Neurosurg Psychiatry 74(5): 558–560. [DOI] [PDF]