SMS is a purely cellular protocol, originally part of the GSM specification for sending small configuration updates to phones, with use for consumer messaging being added a little bit before launch.
So in general it's not possible. In some cases it's possible to receive them without a cellphone (in the UK you can send an SMS to a landline phone, where the network will read it like a machine-generated voicemail. It's annoying and I know nobody who uses this), and you may be able to find a gateway which allows you to send them (a long time ago my then provider used to have an email to SMS gateway, and I'm sure bulk SMS spammers use something like that rather than sending from a cellphone), but a convenient app you can run on an phone and use to send and receive SMS over WiFi, no. That's not what SMS is: it requires a gateway to a mobile carrier's SMS service somewhere.
There are plenty of internet-based messaging services that can be used, provided both parties use that service.