You can buy the vise screw pre-made if it's going to save you a lot of headache, but you can also make a screw of a custom size, in hardwood, by making your own tap and die. The DIY tap and dies work in principle identically to the ones you have in your garrettwade link. And tap and dies for wood operate identically to those used by ironworkers on metal.
With respect to buying a set, the tap and die sets you can buy are generally small, in the 3/8" - 1 1/2" diameter range (at least in the US and Canada). Anything larger would probably require you making one. I can't comment on the relation between the strength of the vise and its screw diameter, but you probably want a screw that will be big enough to withstand dropping a few tools and slabs on it.
The last time I looked, the Lake Erie basic set promoted a maple thread that was approx 2-1/2" diameter and 19-1/4" long, with 2 turns per inch of screw movement.
If you do choose to make your own vise/thread, I can suggest the following resources and ideas.
Option1: make your own tap and die
I was interested about this age-old puzzle, and found a pretty good account of how to craft one, out of virtually nothing but wood, a sharp piece of steel (and a lot of time) in the book:
"Old Ways of Working Wood: The Techniques & Tools Of A Time-Honored Craft", (Revised Edition), by Alex W. Bealer, ISBN 0-7858-0710-1, pp 216.
If we write in terms of "nuts and bolts", the tap will help you form the nut, and the die will help you form the bolt (aka the vise screw). You make the tap first, and then use it to make the die.
I can't replicate the illustrations from the book, but I can explain the gist of it.
You'll need to first find or make a dowel with the desired inner-diameter of your screw. Making a large custom-sized dowel is a separate problem in itself. Short of buying one, you'd have to craft some sort of enlarged blade or chisel housing that works like a giant pencil sharpener. Or you could spin a lathe. Or something like this Matthias Wandel contraption with a drill and a chisel.
Next, you have to choose a thread spacing (i.e. threads-per-inch). There's a bit of a chicken and egg problem in building the tap and die, because you need to model your thread off something. You can resolve this circular problem by neatly coiling a straight strip of paper around your dowel, and drawing a pencil line along the strips, to obtain something that looks like a candy cane / barber pole pattern. Then you saw along the line to make a flat groove. The groove doesn't need to be V-shaped yet.
You can use the flat groove to guide the dowel up and down in a corkscrew fashion, like so: take a block of wood, and drill a hole the size of the tap dowel through it. This gives you a cylindrical hole in which your dowel can slide smoothly up and down. Then, attach a small flat piece of steel (like a washer) solidly to the bottom of the block, offset from the bottom edge of the hole, just so that it peeks out the same distance as the depth of your saw groove. The piece of steel has to be small enough to slide along the groove. As you turn the grooved dowel, the metal piece glides into the groove, and will guide the dowel. So you have a working "corkscrew" motion down and up the cylinder housing.
The next step is to add some teeth along your tap dowel so that you can can use that action to scribe a thread on a nut the same size as your dowel. To make the teeth, it's common to temper the steel of an old file (to make it soft), hacksaw a section off, and then shape the section with a triangular file. It might help to encase a larger section of the file in some sort of mortise in your dowel, and shape out several teeth at once. The point of each tooth should line up with your thread pencil marks. To help with the cutting action, you can make multiple progressively bigger teeth. The largest tooth on your dowel will define the final shape of your groove. Technically you only need one tooth, but more will make tapping easier.
The die is slightly more complicated. You can see an example construction in Youtube: Making Wooden Threads - Homemade Tap and screw box. This is slightly different than what I described above, in that the worker starts from an existing metal screw thread for the tap to make the die box. In this version, the teeth on the tap are formed by filing a cross section of the existing threads.
A single tap box was sufficient in the video example. However, the book warns that it was common for woodworkers to create multiple die box (maybe 3) which would gradually deepen the v-groove of the same thread. So that's something to keep in mind -- it will be time-consuming.
The process in the video seems to mimic the Tap and Die creation described in Andre Jacob Roubo's compendium "Art du Menuisier" (circa 1770), part 3, section 3, chapter 13, plate 311. The original is in 18th century Parisian French, but it has been translated to English in this guide. The method starts with a smooth round forged steel rod of the outer diameter size. If you don't have that, it's still useful because the description provides hints as to how to best design a solid wooden thread (pitch and inclination of thread). The way the initial thread is drawn is also slightly different than what I've described here, so that's worth reading as well.
Option2: make a thread with a router table (and a pantograph)
You can take an existing screw and a router table and customize the thread density with a pantograph. You could then use that screw to make a nut, by adding progressively longer teeth along its thread.
Youtube: How to make wooden threads