Can scratching skin allergies cause permanent skin damage?

Possible, unlikely. Chronic skin conditions that cause itch and subsequent scratching can usually be treated, and if they are treated adequately, the skin can often heal.
Yes. Scratching can cause skin to open and lead to secondary infections. Continued scratching can cause the skin to scar, thicken, or turn darker than your normal skin tone. Best to find out the cause of your scratching and get it treated so you don't cause permanent scarring, thickening.