All citations are from Shulchan Aruch, Orach Chaim 630, except where noted otherwise.
The Theoretically Simplest Idea
The easiest solution, in theory, would be to take a frame, put a post on each of its corners, and you’re done. The Shulchan Aruch says that “some hold” that we apply Gud Asik (extend the roof down to the ground) in such a case (§6). Problem is, the Mishnah Berurah says that we don’t hold of that opinion (§31). So we need a plan B, even though that actually requires some effort.
The Relevant Halachos
Your post makes it sound like you are somewhat familiar with Hilchos Sukkah; I will briefly summarize the relevant halachos for those who are less familiar but also want a portable Sukkah.
A Sukkah officially requires at least three walls, but it’s really two walls plus a tefach (§2-5).
The Mechaber makes a specific note regarding using trees as walls - as long as they’re sturdy enough not to move in an ordinary wind, or they’re tied down sufficiently to accomplish the same, they’re perfectly fine (§10), though there’s a dispute as to how tree branches affect the Sukkah beneath them (SA OC 626:2) that must be taken into account. If you don’t have any trees suitable, remember that the walls can be made from anything, though the Rema limits it to things which will not cause a foul odor (§1).
As you note in your original post, there’s a concept called lavud that we can use as well. Briefly, this means that if parts of the wall are separated by less than three tefachim, it’s still considered a solid wall. This is true regarding a wall being within three tefachim of the ground as well. Further, the pieces of the wall can be as small as you want, and you can apply Lavud however many times you’d like (§9).
One final note: the schach does not have to be directly on the walls; as long as there is ten tefachim of “wall,” the schach can be as high above the walls as you’d like (§9).
Now, let’s have fun combining these.
We start with a basic frame. I might recommend PVC pipe (Ace Hardware has them for pretty cheap). For stability purposes, you might want to make the full box, or at least three walls, but remember that you don’t have to.
Next, take some string. Yes, string. Cheap, easy to find, string. As long as it’s long enough, you can wrap it around just two and a half walls (or maybe three), at just under three tefachim intervals (counting from the top of the base of the frame!) making sure that there’s at least ten tefachim covered in total.
And...that’s it! It has a sufficient number of walls; because of lavud, these minimalistic walls are enough; and the schach doesn’t have to rest on your “wall,” so you don’t need to repeat the strings all the way to the top.
If you have trees handy, you can use them in your frame to lower your cost, too.
This is essentially the same as our theoretical example above, but solves the problem with that one - there’s no walls.
But there’s a catch...
Now, you might notice that this approach, while minimalistic and cheap, doesn’t particularly fit in a minivan trunk easily. (Unless your minivan is actually a U-Haul, in which case I want one.)
You’ll have to find some happy medium here - one that involves enough pieces that it can go into your trunk easily, but not too many that you make it hard on yourself. It really doesn’t matter what the frame is made of, but I think PVC is the best match for your criteria, albeit with a little elbow grease. If you break it up into, say, 2’ pieces, or 3’ pieces, and your Sukkah isn’t particularly tall, you’ll only have about 30 pieces, if that, plus connectors. That sounds like a lot, but it really isn’t; that should take maybe a couple of minutes to put up.
After all, if you wanted it to be super-easy, you’d just buy a pop-up Sukkah online. You asked for DIY - what you gain in money you’ll have to deal with in physical effort.