How about "Bio" vinyl? It is available, for example, as Purline Bio flooring from Wineo. It has already been recommended to us and we have a sample at home that does not smell like plastic at all. According to the manufacturer, it contains oils and other natural raw materials. (But also a bit of plastic, it can't be completely without)
Because of the design, we have now decided on a "normal" vinyl floor from Wineo for the kitchen. The sample we have at home does not have any odor either. In contrast, we had samples of vinyl floors from other manufacturers at home that smelled strongly of plastic and very unhealthy on the underside.