It is very important to understand what we mean by thermal equilibrium when we talk about black body radiation. Thermal equilibrium is meant to say that inside the body, all particles energy level distributions, speed distributions, are the same, and can be characterized by a single value.
That is, its energy level distributions, particle speed distributions (etc.) must be in equilibrium and characterised by a single temperature. Furthermore, the radiation field must also be in equilibrium with the matter at the same single temperature.
Whilst for the interior of the Sun, this is a very good approximation, near the surface it is not, because radiation can escape and the temperature changes with depth on a length scale comparable with the mean free path of the photons. As a result it is better to think of the Sun as emitting blackbody radiation from different layers at different temperatures.
Is sun a black body?
Now you are saying that the sun emits "the radiation coming from the surface of the Sun also follows the blackbody curve" its radiation from the surface, while it seems so to a external observer like us, the photons actually are produced inside the core and make their way through.
Black body radiation is thermal radiation of a hypothetical body that has everywhere the same temperature and is a perfect absorber. Of all bodies at the same temperature, it has the most intense thermal emission at any frequency. The spectrum is smooth, without any lines or peaks or holes, but it drops to zero for zero and infinite frequency as well.
Blackbody radiation Vs thermal radiation
So the answer to your question is, as you see the Sun satisfies all these requirements, it has everywhere the same temperature (at its surface) as seen for an outside observer like us, and the spectrum is smooth, without any peaks or holes.
But if you want to be very specific, you need to think of the Sun as emitting black body radiation at different temperatures from different layers.