My hands and feet are itching with no visible signs of anything to cause it. What's going on?

Do they feel dry? Normally if skin is dry (though it may not look it) it will get itchy due to the dryness. Try to contantly moisture them for 2-3 days nad see if you get any relief. Also use mosituriser without any additives as this can cause more irritation. Use aqueous cream or cold cream.