It has been established that Gamma-Ray Bursts (GRB) can produce Very High Energy radiation (E > 100 GeV), opening a new window on the investigation of particle acceleration and radiation properties in the most energetic domain. We expect that next-generation instruments, such as the Cherenkov Telescope Array (CTA), will mark a huge improvement in their observation. However, constraints on the target visibility and the limited duty cycle of Imaging Atmospheric Cherenkov Telescopes (IACT) reduce their ability to react promptly to transient events and to characterise their general properties. Here we show that an instrument based on the Extensive Air Shower (EAS) array concept, proposed by the Southern Wide Field-of-view Gamma-ray Observatory (SWGO) Collaboration, has promising possibilities to detect and track VHE emission from GRBs. Observations made by the $Fermi$ Large Area Telescope ($Fermi$-LAT) identified some events with a distinct spectral component, extending above 1 GeV or even 10 GeV, which can represent a substantial fraction of the emitted energy and also arise in early stages of the process. Using models based on these properties, we estimate the possibilities that a wide field of view and large effective area ground-based monitoring facility has to probe VHE emission from GRBs. We show that the ability to monitor VHE transients with a nearly continuous scanning of the sky grants an opportunity to access simultaneous electromagnetic counterparts to Multi-Messenger triggers up to cosmological scales, in a way that is not available to IACTs.