Oh boy, I went down a rabbit hole. This has been bothering me to the point where I called in some favors and found some public info for what the current hypothesis on "near" light speed communications is. Because light speed is impossible...for now haha. But let's use some assumptions.
First, light speed travel is possible. Second, that we have superior tracking technology. To the point where we can find the exact location of a traveling at light speed object in space.
The best option we have now is using photonics for communications. This bypasses using radio waves by using an array of lasers that are pointed toward a receiver, which is in turn pointed to the transmitter. We can see why space and light speed travel makes this difficult. It's hard to track objects in space at very high speeds with the precision needed for this level of communication.
The reason photonics is the best option is, it opens up significantly more frequencies to work with. And some frequencies travel farther and faster than others. If we ever master light speed and light speed communications, I would bet money now it's photonics. Because what's faster than light?
Let me know if you have any questions, I'd be happy to answer. Please do not ask about FTL, crazy things start happening then. <- in jest, if you wanna ask you can. Engineers I know argue about it to the point of almost fighting. But that's engineers for you.