Wow! My first big theory! Feels good! I can't wait to see how Book 4 disproves all of this with a single paragraph. So, let's get into things.
Most of this speculation comes from a small argument I had on our lovely Discord about magic systems and methods of access. Yknow, stuff like Bonds and Oaths and breathing in Stormlight. So, at the tail end of this discussion, I had a thought. What is Voidbinding's method of access? Well, you might say Voidlight, but I don't believe the Fused are Voidbinding (yet). But even if that's the case, I assume Voidlight would only be a key, similarly to Stormlight. Not everyone can Surgebind, you need to have a Nahel bond with a Radiant Spren, and you need to say Oaths. So that's what I'm going to call a "method of access" for this theory. With that, let's begin!
The Nightwatcher, the Unmade, and providing external powers
So, two big parts of my theory are these two nuggets of information. First up is the information that Voidbinding usually originates with the Unmade. The second nugget is that Voidbinding and the Old Magic are "cousins", as stated by Khriss in the Ars Arcanum:
So, how does the Old Magic work? As far as we know, it's just the Nightwatcher chucking Investiture at people to facilitate a change. So, if Voidbinding and the Old Magic can be considered "cousins", perhaps the similarities lie here. The Unmade provide an external form of Investiture, either in the form the ability to use Voids/Surges/Voidsurges (terminology unclear), Voidlight, or both. But unlike the Old Magic, Voidbinding seems to be a much more rigid magic system standard of Brandon. It was used (at least in some form, since we can assume that Voidbinding has not been fully explored) frequently enough that the Vorins were able to make a (probably incorrect) chart for it. So, if it's a rigid, Sanderson-brand magic system, how's it work? What's the key? The focus? The method of access? Well...
The empty pit that sucks in emotion
I can't fully confirm what Voidbinding does once it's activated, whether it's Surges used in a crazy new light, to simply corrupted Surgebinding, to something entirely out of left field. But I can ask you a question: why does Surgebinding love bonds so much? I'd think it's because Honor likes bonds and oaths. It's his "thing". Gravitation even works the way it does in Surgebinding because Honor loves bonds. He likes to make them, he sticks by them, and he uses them for all his crazy magic (sealing away Odium, establishing the Oathpact, etc). So what's Odium's "thing"? What does he do? From what we've seen in Oathbringer, and that ominous quote from the Eila Stele I put before this paragraph, Odium "takes away pain". His followers (this is especially apparent in Moash) throw the blame on someone else. They relieve themselves of their pain, and as a result they feel a depressive void. When Dalinar abuses the Thrill, he becomes unable to be sated by anything but Thrill-infused carnage. There is no emotion unless he is accessing Odium's power. While this is obviously a metaphor for drug/alcohol abuse, I do think it still relates to Odium magically.
So, how does this relate to Voidbinding? Well, if we go forward with the idea that the method of access for magic systems is related to the Modus Operandi of their Shard of origin, than perhaps in order to use Voidbinding, you have to drain yourself. Possibly of emotion, possibly of a variety of things that the Unmade ask of you, maybe both, but I believe that Voidbinding requires the user to drain themselves of something. Once they give up their "pain", the Unmade can step in, and provide that internal Investiture source, whatever that may be. I'm not quite sure how it works, but one possible way might be like this:
Some person has Voidlight. They breathe in the Voidlight, and focus on their "pain". The Voidlight gets to work and takes the person's pain. Some Unmade steps in, and forges a quick connection with the person, filling them up with Voidy Juice. Ta-da! You've got yourself a Voidbinder! This might not be exactly how things work, and it might not even need Voidlight at all, but that remains to be seen.
I will give you my sorrow
There's still a chasmfiend in the room: Renarin. He's obviously related in some way to the Void, due to Brandon's chart-pointing and the fact that his Spren is "Enlightened" by Sja-Anat. Since he seems unable to produce illusions (this remains to be seen, but he does seem to have an abnormally difficult time producing them, even accidentally), many people have considered the idea that his Surge of Illumination has been coopted and replaced with the Void/Voidsurge of Illumination. But Voidbinding originates with the Unmade, right? Well, only usually, whatever that means.
So, Renarin bonding a corrupted spren. That might be one of the less common methods of obtaining Voidbinding. Now, according to Brandon, Renarin is "weird' and not really fully Voidbinding. What does he mean by this? Well, there are a lot of theories, but one I think works is that Renarin isn't really doing much - Glys is facilitating everything. It's definitely possible for spren to access Surges. Syl does a bit of it, and Fabrials obviously work that way in some form. And now, this might be a stretch, but consider, Right before Renarin sees his final vision in OB, this little line shows up :
"I will give you my sorrow". And then the next time we see Renarin, boom. Vision time. I might be completely reading too far into this, but if Voidbinding does require giving up "pain" or emotions, than perhaps to use Voidish Illumination, one must give up their sorrow. And since Renarin either doesn't want to or doesn't know how, Glys has been facilitating it for the time being? Can Renarin Voidbind on his own? Does he need an Unmade, or will his strange bond suffice? That remains to be seen.
And with that, all my weird speculation comes to a wrap! Thanks for reading, and feel free to tear into this in the replies. I think it holds some water, and even if I'm mostly wrong, this might be a piece to a larger puzzle. See you all soon!