Does an allergy to Invisalign mean I will not be able to complete treatment?

in case on allergy, primary step is keep yourself away from that allergic agent [in yr case invisalign], so yes , u have to stop invisalign. you can start treatment under allergy specialist's supervision. otherwise you have to stop.

I have had one patient allergic to the Invisalign tray material in 20 years.   H e was truly allergic.   We had to discontinue treatment.   Keep trying and if it gets worse, you may have to discontinue treatment with Invisalign.   Clear Correct is a similar product and uses a different material.   perhaps that might be an alternative.

Allergies are very unpredictable, now you can be slightly swollen and after sometime it might get a very big systemic reaction,  or a disappear, you can never know,  might be a good idea to consult on an immunologist.
If not you are “flipping the coin”