I want to make a project for a client where client can easily weigh a product and get the product weight displayed on his PC or iPhone before submitting the data to a web server, I know some scales have APIs but could they be integrated into the web browser? Or should I do a custom desktop or mobile application to handle that?
If the scales are network capable, then you can write an application using TCP/IP.
If the scales have serial port only, you'll either need a PC connected to it or some embedded device acting like a server. See this device for example, if you want an industrial grade device with rugged case and variable input voltage or you can also take Raspberry Pi or BeagleBone. The server can either provide data over some IP port with defined API or you can implement a web-server there, so that you'll only need a browser on your smartphone (iOS, Android etc.)