Do I need to add studs to install a floating vanity in this layout?

I followed your link to the vendor site, and located a link to a PDF containing installation instructions. It was under Product Page > [Features & Specs] > Technical Specs.

The installation instructions appear to show the cabinet mounting using a "French Cleat" system, with 4 screws through the supporting cleat, and 2 screws through the cabinet after hanging it from the cleat.

In the instructions, the mounting was to plastic hangers, commonly referred to as "drywall anchors". However, the drawings were stylized, so this is not definite.

Since you have studs available, I would suggest that you visit your local home center (or "home centre", eh?) and pick up some hex-headed lag bolts, along with a similar number of matching washers. You can mark the position of your studs during the renovation process (or go back later with a stud finder), then drill larger holes in the mounting cleat to sink the lags into your studs.

Beware, though: my experience with big-box store lags is that many of them are made of "Chinesium." You want to go one or two sizes up from what you think will work. If you're thinking 1/4" (6mm), then go 3/8" instead (9mm).

Also beware: if you and your significant other are likely to "get frisky" on this surface, make sure you look at how it's engineered. I suspect it's probably just melamine (particle-board, MDF), which would contra-indicate friskiness unless you're both "heroin chic".

You do not need to add any studs to the back wall, but I have provided additional hints if you want to make sure.
Attached is a diagram for mounting your floating Vanity. This should be more than enough to have the cabinet and sink be supported on the back wall. Your cabinet comes with a 48” cleat that the cabinet hands off of. The left sister stub or full stub is only needed if the left most stud is not exposed enough to provide a mounting point. Also just use stronger screws ¼” or 5/16” to mount the cleat to the back wall studs. I am guessing the approx. location of where the 48” end on the right side.

The cabinet corner bracket (not shown) is used to prevent the cabinet from lifting off the cleat and is not used to directly support the cabinet. If possible, screw the back of the cabinet to a stub on the right and left side.