Started sulfamethoxazole on tues, today I noticed my lip swelling. Is this an allergic reaction & will the antibiotic still work if im allergic to it?

Probably. It sounds like an allergic type reaction (angioedema). You should stop the drug immediately and call your doctor. Avoid sulfa drugs in the future.
Stop Immediately. If you are having an allergic reaction then it can be life threatening. Stop the medicine immediately and see your doctor asap. They can recommend a different antibiotic which is not related to the sulfamethaxole.
Do not take ! Yes it will work for the time that you took it, but having an allergic reaction is enough reason to stop taking it immediately. Allergic reactions can increase in severity very quickly and can be life threatening. Hopefully you were prescribed this antibiotic by a health care provider & did not self prescribe it. Call your doctor for an alternative rx that will be effective for your situation.