Here are available adapters to connect to web:
Connecting to Internet is longer time ago available also for Commodore 128.
You can select between lot of adapters (shown above) and CONNECT!
Problem is with available programs. Most of them is only for Commodore 64, but the DESKTOP OS Contiki is now available also for Commodore 128, here's link to download where you can choose your configuration:
To use other (C64) webbrowser, you can use Singular Browser, here's link:
http://singularcrew.hu/browser/
To get help how to configure TCP/IP look at this webpage:
http://www.paradroid.net/ipconfig/
Very useful is program WarpCopy to transfer files, you can download it here:
http://www.oxyron.de/html/wc64.html
Also look on webpage about using ethernet on (not only) Commodore 64:
http://home.ica.net/~leifb/commodore/ethernet.html
Want you use your VICE128 emulator with ethernet?
Here you can find solution:
http://vicekb.trikaliotis.net/13-005
In download section is archive with some programs to working with ethernet.
... and look below like interface has twitter client Breadbox64 and webbrowser of OS Contiki...