Merriam Webster does not have "stateful" listed as a word. Within the computer science field, stateful is used as the opposite of stateless:
Stateful means the computer or program keeps track of the state of interaction, usually by setting values in a storage field designated for that purpose. Stateless means there is no record of previous interactions and each interaction request has to be handled based entirely on information that comes with it.
Is there an alternative word or phrase that communicates the concept of something having state? An analogous concept would be software controlling colors of a traffic light, for example, the traffic light software has a state where "the light is green."