Yes, but they would be ... different
This is, conceivably, possible, though I wouldn't try to explain it as a conventional biological organism. Instead, let's focus on directly physics-based lifeforms. In The Five Ages of the Universe, there is a suggestion of emergent intelligence through networked black holes, so there are all kinds of crazy-sounding ideas that could function based on physical principles without traditional biology. Given that we're dealing with stars, light is an obvious candidate.
Our Sun is a fairly average star, so we'll use it for rough approximations to determine feasibility. The basis of this intelligence will be signalling with photons, or groups of photons. The Sun is just over 2 light-seconds across, but light takes about 1 million years to travel from the core to the surface, because it bounces around somewhat randomly for a very long time. Having such long signalling times would slow down the mind immensely, so there are two ways to get around that.
- 'Minor' stellar restructuring: Small tubes of lower-density material could be created (by your gods) to allow photons to traverse the star in reasonable times for a thinking being, but without special constructs, I don't see how these could be held open.
- Quantum entanglement: Though this does not allow for FTL communication, it can allow lightspeed communication without all that finicky structural stuff.
So signalling can be by quantum entanglement of certain photons, but how about the actual computation? It could be distributed throughout the star in computation centers, vaguely similar in function to neurons. This can be achieved through multiple means, including a combination of these:
- Location-based computation: Based on the location of certain photons within the star (which can be determined by interactions with known 'marker' particles), values can be determined and operated upon in various ways.
- Entanglement-based computation: Just like a 'normal' quantum computer, operations can be performed on qubits, and special, non-arithmetic outputs can be obtained, though this would require highly controlled cavitation on small scales to maintain the low temperatures needed.
- Value-based computation: The constants associated with certain particles (ie spin, color, EM field interaction, speed, and mass) could be used to store values around individual ions, operating somewhat more like a conventional computer, though not really through normal electronic means.
The one problem remaining is that this is still just a massive computer system, not a living being. Such problems, though, could be solved just like they are in current AI research. The 'software' running on the stars' computational matrix could simulate a neural network, or some more sophisticated form of learning, to implement a general AI as an independent, living being. Even limbic system (hormonal) changes could be simulated through gradual life changes in the star, and the cycles of luminosity stars normally undergo, as well as random errors introduced by potentially imperfect computations.
Overall, this is an idea that is definitely feasible (and pretty cool) to implement, and I'd love to see where you go with it, even if you don't use my idea!