I accidentally poked a hole in the hose of my waffle iron.

p.s. Every kitchen should have a little fire extinguisher. Write down the simple instructions on how to use on a piece of paper and tape inside a cabinet door. That way you're not trying to operate the extinguisher at the same time as you're trying to read the label on it. And seeing the instructions often whenever you open the cabinet door helps to burn the information into your head.

Phone or email the manufacturer (or check their web site) to see if they will repair it at a cost, even if it is out of warranty. Or if they recommend any particular service vendor in your area. It might cost less to buy a new one than to get this one repaired, especially if the unit is several years old, no longer sold, and so parts may not be widely available. You could also bring it to a vacuum cleaner repair store and see if they will fix it. In any case, be safe rather than sorry. :-)

That "hose" is the electrical cable that carries the current from the lower plate to the upper plate. The "whiteish" color is the heavy duty insulation needed as the cable not only carries the electricity, but builds up a lot of heat.
It is dangerous if not repaired.
Take it to the warranty depot for replacement, or toss it and buy a new waffle maker.
It can very easily start a fire if left unattended to.

There's a whiteish/yellowish hose that's on the back of the waffle iron. I've attached a picture. The right hand picture is from the back of the iron and the left hand ones are from the inside with the iron open. I think you might be able to see where I poked the hole on the bottom left picture.

I think you mean you damaged the wire/electrical connection to your iron? If you wrap it with electrical tape and there are no exposed wires you will be OK.