When we think about what it means to be "conscious", I hope we agree that memory is an essential element. "Information processing" is also an essential element, but many artifacts possess this capability without any long-term memory. For instance, the read/write head of a hard disk may be managed by a PID controller, which clearly has sophisticated information processing capabilities. But I doubt that anyone besides Rupert Sheldrake and his acolytes would say that it is "conscious". Rather, most folks would say that the controller performs its job "mindlessly", not least of which because it requires no long-term memory of its performance, no creativity, no improvisation. Its behaviour, while dynamic at the micro-scale, is fixed at the macro-scale.
Similarly, if you met a robot that could speak conversationally in your first language, but could not remember anything more than 2 minutes old, then you would probably be skeptical of any claims that said robot/AI is "conscious". It may be claimed that it is equivalent to a human who has lost the ability to form long-term memories, but I would argue that such people, tragically, have lost some measure of their "consciousness".
Of course, the brain stores memories in its connections: the synapses between neurons. However, it doesn't store them like bits of data in a computer's Random Access Memory (RAM). The information is stored both by the strength (weight) of the connection, as well as by the topology of the network itself. The where of the connection is just as critical (if not more so) than the how much. And this implies that for the brain to store memories over the long term, it must have a stable topological structure. This is, naturally, why CNS neurons do not, as a general rule, get replaced over your lifetime. New neurons may grow, but they do not "take over" the role of neurons which have died.
The brain is able to maintain its topological structure, because it exists, for the most part, in the solid state. Neurons are slippery, squishy devices, with a significant liquid state component, and the brain itself has the consistency of jello, being able to undergo significant deformations without loss of structure. Even so, if the brain were a fully liquid soup, where any portion could migrate to any location within the volume, then I think we would all be very hard-pressed to explain how it retains memory or function over any meaningful time scale.
The problem with Rupert's argument is that he simply hand-waves away any details by appealing to "structure" and our ignorance, and pretending that putting the two together will lead to consciousness. But the big problem you have with the sun is that it is not solid. Nor is it semi-solid, like the brain. Nor is it a liquid, like the ocean. No, the sun, from its surface down to its core, is one blazing-hot plasma. And anyone who has tried to build a plasma-based fusion reactor knows that a plasma that is hot enough to fuse H, D, or T is so unstable that there is nothing like a long-term structure capable of holding a memory.
When we look at the environmental operating range of the brain, we see that it is a very fragile instrument. If it goes just 10 C above nominal, it can seize up and stop functioning entirely. While silicon processors have a much broader operating range, they are also much more robust by being literal crystals of metal. The intra-molecular bonds of silicon chips help them retain their function over a much broader temperature range. The sun has an internal operating temperature of about 15 million K. I think it is quite safe to guess that the spatial relationship of any two atoms within the sun does not survive long time scales (for just about any definition of "long time scale").
Now, this is again hand-waved away by claiming that consciousness is not, in fact, stored in the structure of the brain, but rather in the electromagnetic fields. This is just silliness. If having an "electromagnetic field" is sufficient to have consciousness, then everything (made of normal matter) has consciousness, including my coffee cup. And at this point, "consciousness" ceases to be a concept, because it is equivalent to "vwerp". You may never have heard of vwerp before (in fact, I really hope you have not) because I just made it up. It doesn't matter how you detect vwerp or whether it's interesting or harmful or anything else, because the one thing you need to know about vwerp is that every object has vwerp. And once you know that, you know it is an utterly useless and meaningless concept. Well, you also need to know that there is no way to distinguish the vwerp of one thing from that of another, because the vwerp is ultimately shared amongst all things. So it's not like position or momentum or other physical properties that we presume everything also has. It's universal and indistinguishable.
This clumsy sleight of hand is necessary to remove the topological structure of the brain as a necessary requirement for other conscious artifacts. It is the most obvious thing which makes brains different from coffee cups and clouds and ironing boards. But all of those have electromagnetic fields of some sort, so they also all have consciousness, according to Rupert. And that leads us to the Final Silliness.
Rupert is not really interested in physics, so asking about the physics of the sun is really missing the point. He believes in a magic phenomenon called "morphic resonance". You can test this phenomenon by recruiting a friend, and having them stare at you when you are not looking. You then record when you sense being stared at, and when your guess is correct, you have confirmed the theory of morphic resonance! This theory has, in fact, been validated thousands of times...on the internet...by volunteers. It's science, right? If you think so, Michael Shermer would like to have a word with you. You see, some people...whom you might call "real scientists"...have also attempted to replicate the morphic resonance experiment, and failed. But, not to fear! Those scientists suppressed the morphic field with their doubt. So, in fact, it is impossible to falsify morphic resonance, because any attempt to do so will destroy the very phenomenon being studied!
So you see, it doesn't matter if the sun has large-scale structure or not. It doesn't matter whether plasma granules behave like silicon granules or whether the magnetic fields in sunspots are strong enough to store memories. Because at the end of the day, the morphic field is not about science. It's like asking whether a salty lake is an adequate test environment for detecting witches.