I believe it's possible to have irritation (e.g. contact dermatitis) caused by some of the components of some plastics, but that's unlikely to be the focus of your allergen program. I assume you're manufacturing plastics for use as food packaging?
If so, your allergen program will primarily be about preventing accidental contamination of the product with food allergens. For example, if one of your colleagues eats some peanuts on their break and then starts handling product without washing their hands, putting their PPE on etc, they are potentially going to transfer proteins from the peanuts onto the packaging, and from their it can get transferred onto whatever food product your customer packs in it, posing a risk to consumers who are allergic to peanut.
Your risk assessment is therefore likely to primarily focus on accidental routes of potential contamination with food allergens.