You can't access a SIM card that way, but a SIM card usually won't have your phone numbers. That is just to give the phone service through whatever carrier you are using. If you mean an SD card, the info wouldn't be on there either, unless you saved it to the SD card intentionally. Contacts and other data are normally saved on the phone itself. To the phones storage, which isn't on a SIM. So the SIM won't have what you need.
You would need to either find a way to access the files on the phone, or check your phone bill (paper or online) to see the calls you have made and received. As well as texts sent and received. That would give you the numbers, but no names or other data.