Itching of the palm and fingers is most often from
hand eczema or contact dermatitis: irritation or allergy from soaps, sanitizer, detergents, gloves, cosmetics, metals such as nickel, or frequent wet work. Hand eczema can also occur as
dyshidrotic eczema, which causes very itchy tiny blisters on the sides of fingers or palm.
Andrews' Diseases of the Skin lists irritant dermatitis, allergic contact dermatitis, atopic eczema, and recurrent vesicular eczema among the common hand-eczema patterns (p. 3881-3898).
Mayo Clinic describes dyshidrosis as itchy fluid-filled blisters on palms and sides of fingers.
Other possibilities include dry skin, a fungal infection if there is scaling mainly on one hand, or scabies if itching is intense at night and there are bumps between fingers or other household members are itchy.
Seek urgent care now if your hand, lips, face, or throat swell, you have breathing difficulty, rapidly spreading hives, severe pain, fever, pus, or red streaks. Arrange a clinician or dermatologist visit if it lasts over 1 to 2 weeks, keeps returning, or you develop blisters, cracking, or a spreading rash.
NHS information on pompholyx notes that this type of eczema typically causes recurring itchy hand blisters.