We show that a III-V semiconductor vertical external-cavity surface-emitting laser (VECSEL) can be engineered to generate light with a customizable spatiotemporal structure. Temporal control is achieved through the emission of temporal localized structures (TLSs), a particular mode-locking regime that allows individual addressing of the pulses traveling back and forth in the cavity. The spatial profile control relies on a degenerate external cavity, and it is implemented due to an absorptive mask deposited onto the gain mirror that limits the positive net gain within two circular spots in the transverse section of the VECSEL. We show that each spot emits spatially uncorrelated TLSs. Hence, the spatiotemporal structure of the light emitted can be shaped by individually addressing the pulses emitted by each spot. Because the maximum number of pulses circulating in the cavity and the number of positive net-gain spots in the VECSEL can be increased straightforwardly, this result is a proof of concept of a laser platform capable of handling light states of scalable complexity. We discuss applications to three-dimensional all-optical buffers and to multiplexing of frequency combs that share the same laser cavity.