This page of the guide to Mafia Definitive Edition will tell you where to find the APC in the game, similar to the one that chased you during one of the main missions.
Yes, the armored vehicle is available in the game and it is very similar to the vehicle used by the lawmen in the story mission Trip to the Countryside.
You can find the armored vehicle in the location shown in the picture above - this is the Federal Post Depot building located at the northern end of the North Park district of Lost Heaven. If you don't see this machine parked in front of the building, try to return to it later in the game. You can simply enter the APC – it doesn't have a locked door.