My granddaughter is allergic to rubber/latex. What would be an alternative to rubber handles on her bike.?

Visit your bike shop. Handlebars grips are available in plastic and other non-latex materials. Leather gloves will prevent direct contact with rubber grips as well as cotton does while providing additional hand protection from falls.
Wear cotton gloves. ...If needed. True latex allergy is in reaction to airborne latex rubber proteins. Vulcanized rubber, such as that in bike handles, tennis shoes, rubber bands, etc, does not give off latex proteins. Many people allergic to latex are afraid to be around this type of rubber, but it really doesn't cause any trouble.