Texting over WiFi is NOT an app-dependent feature, unfortunately. If your carrier doesn't support it, the app you use won't either. Apps like Google Messages (free) and Pulse (paid) can let you use cloud features to text from another device/browser, but they all require the originating phone to be online (and, again, whether or not it can send/receive texts via WiFi or only via carrier signal, as mentioned before).
If you're looking for a messaging app (not SMS/MMS) that can work both on WiFi, mobile, and across devices, then you have a plethora of options, like Facebook Messenger, Whatsapp, and Telegram (but these all require both sender and receiver to have the app as well).