There is a service station on the A14, called Daunia. You can get there by walking east from the train station, towards the Via Leone XIII which will then become Via del Mare. It's a bit of a walk until you get to the Statale 673, but this is not the motorway yet! It's the second big street you come across. If you want to go south towards Bari, turn right before crossing over the motorway, if you want to go north towards Pescara, turn right after the motorway. Both service stations can be accessed by the country road the staff usually takes. The fences are not very high and can be climbed, but also check if the doors are just open.
(I was here on 30/03/2016 and I believe these service stations were closed. Couldn't arrive to the one I wanted -direction south to Bari- because the country road was blocked with a sort of barricade. On the distance I could see the service station's sign was covered with plastic and I'm pretty sure it was empty. From the bridge that crosses the highway I saw that the opposing station -direction north- also seemed empty and closed.)