Uh, Tom and Tom, I sure would not. Sheet vinyl linoleum flooring will not absorb water or let water pass through it. In a kitchen, you will have water spilled. Water hurts wood. The vinyl is a much better and safer option.
Granted, you and your family might really take care of the wood floor in the kitchen and never have a problem. But some day you will want to sell the place and the lack of traditional materials will hurt resale value. Ceramic tile is a better "natural" option, but my preference for easy maintenance is still the vinyl.
Something will leak--refrigerator, dishwasher, sink or other plumbing--or someone will spill something like a gallon of milk and the odor will stay with you for generations. My opinion of wood floors, including engineered wood, in the kitchen or bathroom is still, yes, you can, but NO, you shouldn't.