If you've used the Thingiverse design to cut your pieces out of wood, you should have the parts shown here: Candle pieces 1, 2, and 3, plus a base and a switch arm.
1 LED. I like the jumbo 10mm LEDS, for example this yellow one from Jameco.com
1 CR2032 battery, available from adafruit.com
and at most DIY hardware stores
1/4" adhesive copper tape, you can find it on amazon
(see below for alternatives)
A bamboo skewer or short 1/8" wooden dowel
A strong clamp (an office binder clamp works well)
What if you don't have a laser cutter?
You can cut these pieces out of 1/8" corrugated cardboard using a craft knife, or out of 1/8" wood using a scroll saw. You can even cut them out in layers of thin cardboard with scissors, and glue the layers together. if you choose a different material just pay close attention to the thickness of piece #2: It should be almost the same thickness as the battery. All of the parts need to be strong enough to hold up to use, and they need to be non-conductive, but if you keep those things in mind you can cut them out of anything you like.
Alternatives to copper tape
Since I was making a lot of these at Ace Monster Toys I bought a whole roll of 1/4" adhesive copper tape, but that's expensive. One alternative is to buy a smaller roll of copper tape from your local garden center - they sell it there as a snail deterrent. You'll need to cut it down into 1/4" strips. The simplest alternative of all is to use ordinary aluminum foil from your kitchen. Again, you'll need to cut it into strips, and you'll need to glue it. Rubber cement will work better for this than wood glue. Aluminum foil is thinner and more delicate than copper foil, so the key thing is not to leave any jagged edges that could tear as you move the switch arm.