ビニール is the transliteration of the English word “vinyl” but what it means is not the same as “vinyl”.
Any plastic bag you can get in a super market to put vegetables in is called ビニール袋（ぶくろ）. A plastic umbrella you can buy at a 100-yen shop is called ビニール傘（がさ）. A blue plastic sheet that covers a damaged roof is called ビニールシート.
In short, almost all soft plastic made into a thin film/sheet is called ビニール in Japan. One exception to this I can think of straight away is “cling wrap” for food, which is usually called 食品用（しょくひんよう）ラップ or simply ラップ, but other than that, most soft plastic is called ビニール regardless of the actual material.
The Japanese word プラスチック only refers to hard plastic.