It's possible since at least 2 years, maybe more. You still need a smartphone to create the account, but apparently you don't need the phone to be online, at least not all the time, not sure if it keeps working if it's offline for months. (note I'm not a Whatsapp user, so maybe it changed again or I didn't fully see something...)
you don't actually need a smartphone at all. i discovered this by accident when my smartphone died. here is what i did.
- installed the whatsapp "app" on my crappy old android tablet that i hardly ever use
- used SMS to my dumbphone to link the installed app to my phone number
- opened "web.whatsapp.com" on my laptop browser
- scanned the QR code in the android app to link the browser session to my account
so, i am now happily using web.whatsapp.com in my laptop browser day to day and all i needed was a crappy android tablet and a dumbphone to set it up.
i have been using web.whatsapp.com on laptop for long time - pretty sure it's 3-4 years at this stage. maybe i am wrong on that.
i also find the browser/laptop whatsapp much nicer to use - copying/pasting etc. is so much easier than a smartphone, for me at least.
