Yes both my Kobo and my phone connect to the internet and I am able to browse everything using the Kobo browser including the Kobo website. I can even search the Kobo store from my e-reader but it just won't sync.
I figured it might be something router related but I can't figure out what. Sadly Netgear don't allow you to tinker with the firewall, but then again the PC and BlueStacks can access their respected stores due to being wired...