You have two easy and familiar routs where you could:
1. Biolife in the blood
This would be largely science fiction, but it would have "believability" for your audience. Some kind of proactive lifeform in the blood, such as a bacterium, would fight to destroy the virus. Here, the blood is just the carrier and, while a transfusion could work, isolating the thing inside would be more efficient, allowing it to be distributed on a mass scale more easily.
That could use "option exhaustion" to propel the plot and give you the scientific breakthrough needed for victory, either for the protagonist or a helper.
Something similar was done in Deep Space Nine, Season 1 (ep13 'Battle Lines'), where the spiritual leader Kai Opaka died on a remote moon, but was brought back to life by nanites in the atmosphere. Nano-tech is a stretch, but bacteria is more believable and adaptable.
Vaccines are basically made from dead viruses. The blood of someone immune would probably have traces of it, or antibodies for the virus. But, it may need engineering and would require blood type matching. A transfusion itself probably would't work very often if at all, the plot would keep waiting for a scientific analysis of the blood to make a vaccine.
This is similar to what started at the end of I am Legend with Will Smith. Although that blood sample did contain a kind of vaccine he had introduced into the blood.