On the side of a package if it states "Cholesterol 0%" in the Nutrition Facts does that mean the food is vegan?
No. The per-serving nutrition numbers are rounded and only reflect the value for a single serving. A value of 0 simply means "less than 0.5 mg" in a single serving.
Your question implies that cholesterol only comes from animal products. This is not correct. Cholesterol is present in many plants.
Other answers and comments claim that only amounts "less than 0.5" (units omitted) of cholesterol is permitted to be listed as 0, and that "no cholesterol" is an added claim that a product is truly cholesterol free. This too is incorrect. The FDA permits amounts less than 2mg/serving to be listed as both 0 and "cholesterol free". There are additional caveats which can be read at the link.
The bottom line is, not only does 0 mg of cholesterol not indicate anything regarding the presence of animal products, but neither does the presence of cholesterol.
Another point to mention is that there are many animal products that have no cholesterol. A great example would be honey, or egg whites. (Or gelatin, I believe) So even if you could actively figure out whether there is any cholesterol, that would not mean that there are no animal products in the food.
As mentioned by Joe and gordoco, US labels aren't of much help for this level of detail due to the ridiculous rounding rules. But there are other sources of info. Check out Calorie Count for example. That site's info is accurate down to .1 probably due to its origins as a european-based web resource. (Europe's governments seems to trust its public with decimal notation.)
Still, even though it's a much more accurate source of info you shouldn't count on it alone to decide whether the product is vegan. And depending on how strict you want to get even a food that is labeled as entirely animal product free may have been processed in a very un-vegan manner. The discussion of whether your basic Heinz Ketchup is vegan is a prime example. (Bone char is used in the processing of the sugar which is used in the final mixture.)
Almost all organic vegetables are not vegan as they use Blood Meal fertilizer. It is made of dried and powdered blood of animals (usually cattle), and is used as a nitrogen-rich fertilizer for plants. It has been utilized by organic gardeners for years. Ironically, many vegans choose to be vegan for moral reasons.