Hey Holly! That's a nice site! You could create something *kinda* similar in Articulate Presenter... here's a super-simple mockup that I threw together quickly.
There are a couple challenges with a vertical sliding menu like this... one is the fluidity and "smoothing" effect that you see in the website example - where the vertical menu bars slide into place from one side or another, and they kind of slow down before they stop moving. Unfortunately that slow-down isn't possible to replicate in content that you publish from PowerPoint to Presenter.
Another challenge is the direction of the sliding motion. There isn't really a way in PowerPoint/Preseneter to predict which slide a user will be coming from when they arrive at any particular slide - which means you can't really apply intelligence that tells the animation which direction to start from. (Wow that sentence was really a grammatic mess, lol!)
So, those are the reasons why in my mockup I just used a fade animation on each of the content slides. It's not horrible, but it's not as slick as the example you posted.