Having the SIM really has no effect on access to contacts.
If your old phone was connected to (and using auto sync) Google, then you may find your contacts on there. With that you can copy them to the phone. However, if they weren't there, or saved somewhere else by you, then you would have lost them.
Contacts are saved on the device, not the SIM or on an SD card.
The best you could do is see if you can check your Call and Text history through your carrier/service provider and obtain numbers you have called, gotten calls from, texted or gotten texts from. But that info will only go so far back, and won't have names attached.
In the future I would recommend saving your contacts information to a computer. The contacts on an android gives you the option to export the information, which you can then save to a computer, cloud account, etc.