Client and server model in a web game

Currently, I am developing a 2-player tetris game. Originally, I was planning to implement a client-server model in a mobile environment.

However, now I change into web game, which is accessible by a browser. Since I am not used to a web development, I want to know if client-server model is necessary in a web game.

To be honest, I am a little bit skeptical writing both client and server code since a browser does a job to connect to the server.

Is it possible to build a network-based game by using only one server without a client part?