Hi - welcome to adafruit!
The answer is yes, you can write to the sim. Here's what it says in the product description:
This is a SIM card reader/writer kit for experimentation and investigation of SIM & Smart cards. Once the kit is assenbled, accompanying software can be used to read and write from the card. Together they can be used to backup stored SIM card data, recover deleted SMS's and phone contacts, examine the last 10 phone numbers dialed, etc. (Despite being called a SIM reader, it can also write to SIM cards)