What is the sticky stuff coming out of my wood? (there must be a better way to have said that...)  

hot wet secretions from naked human flesh
sweaty oily malodorous..."patina"
the filth of the human animal
rejoice in it
rejoice

Were you using some sort of detergent or cleaning product? Did the surface remain sticky after it dried? Sometimes when vinyl breaks down, plasticizers can leach out and off-gas a "cheesy" odor. I treat vinyl with "303 Aerospace Protectant", though I'm not sure that it would help with the offensive smell.

Nicotine and cig tar leave a yellowish brown film so it's probably not that, unless they have reacted with your cleaning agent and the finish and it's all coming off...?? I kind of doubt that, though.
Is there a finish on the wood? It would probably be helpful to know what it is so that you can use the right stuff to clean it. I would test for lacquer first---dip a Qtip in nail polish removier and dab it in an inconspicuous spot. Lacquer will start to dissolve within 5-10 seconds and the Qtip will come up brownish at the very least.
If that doesn't do it, you could test for shellac the same way with alcohol, but i doubt it's shellac.
If it's varnish, neither of those two solvents will dissolve it. I've refinished a few Plycraft chairs and they were varnished. I had to use mehtylene chloride stripper on them. Yech.
But anyway, you're not refinishing. What cleaning agent are you using that turned the finish sticky?
Vinyl can absorb bad odors and hang onto them for a long, long time, unfortunately. I just packed a vinyl doll reeking of cigs in baking soda for a week---didn't even touch the odor. Next I tried soaking it for a day in laundry detergent and Oxyclean, and now it smells like laundry detergent and Oxyclean with undertones of cigs. Yum.