An HDMI socket cannot be used to acess Wi-Fi. This facility is either built into equipment or provided by insering a USB dongle into a USB socket specially provided for this purpose. TVs which have such capability are described as "Smart" and have only fairly recently been introduced. Hence it is almost certain that your current TV cannot accept a Wi-Fi signal.
The internet access provided by Smart TVs or other devices such as DVD players with wi-Fi tends to be limited to various entertainment sites such as Netflix, Lovefilm, YouTube etc., although newer models are expanding the capability. I believe that Sony make a TV with the most comprehensive capability.
One way to get internet to your existing TV without using it as a monitor for a PC or laptop would be to use a Sony NSZ-GS7 Internet Player link. This accesses the internet via a wireless router, which I presume you already have. I realise that this is a fairly expensive product but I am not aware of an alternative which meets your needs, as I understand them, apart from accessing the web via a PC/laptop/tablet and using it your TV as a monitor.
I should add that I have no connection with Sony and offer the foregoing based on my somewhat limited knowledge the latest technology.