We have observed the late-type peculiar galaxy NGC 4424 during VESTIGE, a blind narrow-band Halpha[NII] imaging survey of the Virgo cluster carried out with MegaCam at the CFHT. The presence of a 110 kpc long HI tail in the S direction indicates that this galaxy is undergoing a ram pressure stripping event. The deep narrow-band image revealed the presence of a low surface brightness ionised gas tail ~10 kpc long extending from the centre of the galaxy to the NW direction, in the direction opposite to the HI tail. Chandra and XMM X-rays data do not show any compact source in the nucleus nor the presence of an extended tail of hot gas, while MUSE spectroscopy indicates that the gas is photo-ionised in the inner regions and shock-ionised in the outer parts. IFU spectroscopy confirms that the ionised gas is kinematically decoupled from the stellar component and indicates the presence of two kinematically distinct structures in the stellar disc. The analysis of the SED of the galaxy indicates that the activity of star formation has been totally quenched in the outer disc ~ 250-280 Myr ago, while only reduced by ~80% in the central regions. All this observational evidence suggests that NGC 4424 is the remnant of an unequal-mass merger occurred <= 500 Myr ago, when the galaxy was already a member of the Virgo cluster, now undergoing a ram pressure stripping event which has removed the gas and quenched the activity of star formation in the outer disc. The tail of ionised gas probably results from the outflow produced by a central starburst fed by the collapse of gas induced by the merging episode. This outflow is sufficiently powerful to overcome the ram pressure induced by the intracluster medium on the disc of the galaxy crossing the cluster. This analysis thus suggests that feedback can participate in the quenching process of galaxies in high-density regions.