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According to their website, Skippy brand peanut butter does not contain trans fat.

However, the current label for their chunky peanut butter contains the following ingredient in the listing: Hydrogenated Vegetable Oil (Cottonseed, Soybean and Rapeseed Oil) To Prevent Separation.

Soybean oil is a known source of trans fat, which can be a source of atherosclerosis – the clogging of arteries with cholesterol – which can be a cause of heart attacks.

Is their website just blatantly lying about the lack of trans fat?

According to their website, Skippy brand peanut butter does not contain trans fat.

Strictly speaking, according to their website, Skippy brand peanut butter contains a negligible amount of trans fat. Per the US FDA,

The Nutrition Facts Label can state 0 g of trans fat if the food product contains less than 0.5 g of trans fat per serving. Thus, if a product contains partially hydrogenated oils, then it might contain small amounts of trans fat even if the label says 0 g of trans fat.

So they're being sneaky, right? Not really. The amount of trans fat in a serving of peanut butter is far less than the limit of 0.5 grams that needs to be reported as above zero. Only a tiny amount of hydrogenated vegetable oil is added to peanut butter to make it smooth, prevent separation, and drastically increase shelf life, and only a small amount of that small amount is in the form of trans fat. While non-zero, the amount is essentially undetectable. From Sanders, T.H., 2001. Non-detectable levels of trans-fatty acids in peanut butter. Journal of agricultural and food chemistry, 49(5), pp.2349-2351,

The fatty acid composition of 11 brands of peanut butter and paste freshly prepared from roasted peanuts was analyzed with emphasis on isomeric trans-fatty acids. No trans-fatty acids were detected in any of the samples in an analytical system with a detection threshold of 0.01% of the sample weight. Hydrogenated vegetable oils are added to peanut butters at levels of 1--2% to prevent oil separation. Some hydrogenated vegetable oils are known to be sources of trans-fatty acids in the human diet. The addition of these products was not found to result in measurable amounts of trans-fatty acids in the peanut butters analyzed.

I note that the label says "Hydrogenated Vegetable Oil" not "Partially Hydrogenated Vegetable Oil". If the oil has been fully hydrogenated it no longer contains trans fats, by definition.

Unfortunately you can't know whether an oil has been fully or partially hydrogenated when the label omits those qualifiers. It seems highly likely to me that the manufacturer did this intentionally to avoid the negative stigma associated with partial hydrogenation. It would be wise to assume partial hydrogenation when "Fully" is absent.

A fully hydrogenated oil by itself will be solid at room temperature but I cannot say what would happen to an fully hydrogenated oil suspended in a matrix of peanut bits.

http://www.berkeleywellness.com/healthy-eating/food/article/hydrogenated-oils for more information on hydrogenation and health.

https://www.nationalpeanutboard.org/wellness/why-you-shouldnt-be-scared-oil-in-your-peanut-butter.htm for a highly biased take on the oils used in peanut butter across the industry.

The FDA has determined that Partially Hydrogenated Oils are not Generally Recognized as Safe (or GRAS) and has established rules for their elimination from the US food supply:

https://www.fda.gov/food/ingredientspackaginglabeling/foodadditivesingredients/ucm449162.htm

If any food still contains trans-fat, it won't for long; they are being from all foods.