Laser-accelerated electron beams, in the so-called Very High-Energy Electron (VHEE) energy range, are of great interest for biomedical applications. For instance, laser-driven VHEE beams are envisaged to offer suitable compact accelerators for the promising field of FLASH radiotherapy. Radiobiology experiments carried out using laser-driven beams require the real-time knowledge of the dose delivered to the sample. We have developed an online dose monitoring procedure, using an Integrating Current Transformer (ICT) coupled to a suitable collimator, that allows the estimation of the delivered dose on a shot-to-shot basis under suitable assumptions. The cross-calibration of the measured charge with standard offline dosimetry measurements carried out with RadioChromic Films (RCFs) is discussed, demonstrating excellent correlation between the two measurements.