It really depends on the grill itself and how it works. Several of them only support iPhones and Android phones for their WiFi. I think I have seen one or two that say they work with any desktop browser as well. However, you most likely won't get alerts from that setup.
I mean absolutely nothing by saying this, but do want to make sure you know that you can control all of the WiFi grills I've see from the grill itself...it doesn't require WiFi to run. Supporting the idea of someone running with a desktop browser would be relatively simple if you are already writing apps for phones, but I can see where is doesn't have much traction and justification to the grill companies. Most people want WiFi so they can check on their grill and get app alerts or text messages when their food is done and they're busy running errands. If you're at home on your desktop, you'll probably just look at your grill (in their minds).