So, does the Vaccine provide Immunity?
Yes.
Immunity in medical terms is not complete protection from the virus, it just means your body has the ability to fight the virus.
Almost all the current vaccines have been proven to reduce severity of symptoms of Covid by a factor of 90% reducing the death rate significantly among affected.
Due to the high replication rate of the virus, especially delta strain, live virus can still be shed and yes a person may still have active infection along with being infectious to others.