Hi Aileenfromeden,
I don't have any experience with CarPlay, but I'd be more than happy to hopefully help us get this figured out together! To start off, If I'm right you'd like to see your contacts on the display in your vehicle. The only problem is that your contacts are only displayed when the iPhone is connecting to the car through bluetooth.
I may be wrong but my best guess would be the car may probably not have some form of "storage" media so it can't store your contacts, which might explain why you're not able to see your contacts unless you connect the device to car with bluetooth. From what I've seen around the web though, If your car has an iPhone plug, you could try connecting your device to the plug inside that car, and CarPlay should automatically recognize it. Once CarPlay recognizes the device, see if you can access your contacts. If you can, then the next time you can just leave your iOS device plugged in, and not have to use bluetooth.
I'd also recommend checking out this support article from Apple about setting up CarPlay with your iOS device just for other resources if you wanted or needed more information, which is located here: Set up CarPlay to use with your iPhone - Apple Support.
I hope I was able to help you, or at least provide you some information. If you have any more questions please don't hesitate to ask.
Regards,
John