Ping between a Chat application and SMS

I am thinking of developing a chat module in a web application where a customer wants to chat to the administrator through the chat module. The twist here is the message the customer shoots from the chat application should to be sent to the phone of the administrator through SMS and he/she could be able to reply to the customer back as a SMS reply which will be responded back to the chat module of the web application. Am I sounding reasonable? If so please guide with some start steps to achieve this.