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Hi Soulman!
I found this video, using sketchup: https://www.youtube.com/watch?v=0JSkczTHAh8
Sketchup isn't the best program for 3D printing, but the idea is there, you can use any 3D program to do the same process.
1- import the vector file (normally as dxf)
2- select it and extrude it to the the thickness you want it to be. Note that in some programs like Sketchup you need to close the 2D shape first.
3- save the 3D file, compile it in G-code, and 3D print it! :)

in most laser-cut projects (not this one) you'll need to assemble several pieces, like a puzzle. This implies fits in the pieces in order to assemble them together.
This fits are normally designed with a specific material thickness in mind. For example, my designs here on Thingiverse are all designed for a 3mm thickness material. If I wanted to laser-cut the same project in 5mm, I would have to
1) scale up the project dxf 166,(6)%, so that the fits increase from 3mm to 5mm, and the resulting final object will also increase in size accordingly; or
2) if I want the final object to keep the same size, I would have to adjust each fit individually in the dxf and transform 3mm fits in 5mm ones (there may be some fits that you won't need to adjust, it depends on it's specific design. But most of them will probably need adjusting and it's normally a simple thing to do, just a bit tedious :\ )

As an example for this second point see my Portable Chess Set files - https://www.thingiverse.com/thing:1720044/#files
There you can see the original file, designed for 3mm(1/8in)
and an adaptation for 1/4in, requested by another user (krickard8)
(don't know if the adaptation works x)

With this leafs, don't worry about it. There are no fits to worry about so you can print it with the thickness you wish (there is always the resistance/aesthetic relation to worry about: the thinner it is, more fragile it becomes; the thicker, more ugly it gets).

also important:
when importing/exporting a dxf file between programs, keep the measures in check. It's not uncommon to import a dxf draw in another program and find the measures all scaled down. This happens mostly due to different unit set ups between programs. For example, I use metric system. If you import any of my files with a program set in the imperial system, the measures of the dxf will probably be scaled incorrectly. All you'll need to do is scale the draw back to the desired measures or change your units and import the file again. This solves most of this scale errors.

Hope this helps! :)

Portable Chess Set