My daughter has an iPhone 5S. She has an unlimited text plan only. She isn't receiving texts regularly when she isn't at home on wifi. I'll text her to make arrangements but won't get the text until she is on wifi again. It's very unreliable.
She likely has Messages turned on on her phone. Messages needs data, either cellular or wifi to send and receive information. This would be why she only receives the texts you send until she connects to wifi. Her options are to turn off Messages in settings, get a data plan to use Messages, or use it the way it currently operates. Turning off Messages will cause the phone to use SMS instead, but others who also use iPhone Messages to contact her may have a period where their phones have to find her again on SMS.