She calls them whenever she is travelling to assure them she is alright.
Similar ‘merged’ words such as altogether and already have been accepted in standard English for a very long time, so there is no logical reason to object to the one-word form alright. Nevertheless, many people dislike it and regard it as incorrect, so it’s best to avoid using alright in formal writing. Write it as two separate words instead:
She calls them whenever she is travelling to assure them she is all right.