I am Gregg with The Home Depot in Chicago. With that amount of weight hanging from the wall you are going to want it distributed as equally as possible. One way is to install a 2x3 against the wall for the vanity to rest on and put 2 screws in every stud along the 129” span. I would also put toggle bolts in between each stud for extra support. You could even add an angled support from the bottom of the vanity to the 2x3.
Another way is to use angled iron to connect to the wall. Install a 2x3 along the wall the entire 129” span at the top of the vanity and a 2x8 along the bottom of the vanity. Make sure the 2x8 goes 2 ½” past the bottom of the vanity so you can add an addition 2x3 on top of it. Add angle iron to the top of the vanity and the bottom of the top 2x3 so the two pieces will hook into each other. Rest the vanity on the bottom 2x3 that is over the 2x8 and add additional screws. This way would push it out from the wall 1 ½”, but since you are custom building the vanity you can account for the difference and cover the sides.
let me know if this helps.
(The pictures are viewed from the side.)