As Thisjrp4 said vinyl is a very poor conductor of heat and will not exchange the heat as the wort passes through it. As many others have said the vinyl will not stand up to that temperature. Finally a small lunch style cooler does not hold even close to enough ice even if vinly did exchange heat. When you first start to cool the wort, that 200 + degree wort will burn through ice rather quickly.
If you want to make an immersion chiller the tubing really needs to be copper which is an excellent heat exchanger at a reasonable cost. If you need to do it on the cheap you might try watching craigslist for someone doing plumbing work who has some copper pipe left over. Otherwise one of the big box hardware stores and 3/8" copper tubing is the best bang for the buck. 25' worth isn't that bad in price. I think I bought a 40 or 50 foot coil for like $35
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