Is it possible that a child suddenly shows signs of allergies for the first time, at almost 5 years of age?

YES. Allergies happen when your bodies immune system become "sensitized" to something "x" and develop antibodies to it. Over time your child has multiple exposures to x (years 1-4) and became sensitized. We are rarely if ever allergic to anything when born; but develop them over time (and exposure). So it would be quite normal to "suddenly" become allergic at age 5 even if they had it many previous.