Yes, you can have eyelash extension, but you need to be very careful.
For sensitive eyes like you, the operator can't be more careful enough to avoid allergy or irritation. They need to prevent from touching your skin with the false eyelashes.
Also, suitable glue should be chosen with great attention to keep allergen away from your eyes. Natural ingredients have less risk of infection or irritation.
In addition, make sure the operator have his or her work done properly. All the tools should be disinfected thoroughly and appropriately.
All in all, the more attention you have paid, the less risk of irritation will you have.