The hit and run followed angry scenes on adjoining Pershore Road after men from the Raise the Colours group arrived to put up flags.
Videos shared on social media show arguments between the group members and local residents.
West Midlands Police recovered the vehicle and said it had been reviewing CCTV footage and videos on social media.
Birmingham Selly Oak MP Al Carns wrote to the acting chief constable, saying he was “seriously concerned” by events in Stirchley and that residents “deserve to feel safe in their own neighbourhood”.
The city council previously said it was aware of the incident, and “increased tensions in some areas”, and encouraged calm and “community cohesion”.