This needs a two-part answer, because the oil and the capsule shell come from different sources.
The oil inside is entirely plant-based: flaxseed is the plant source, and this is exactly why the manufacturer lists vegetarian and vegan diets among the typical cases where people choose flaxseed oil, together with periods of restricted or monotonous eating, and a wish to add more plant-based unsaturated fats to the diet. The fatty acids in the oil (ALA omega-3, linoleic acid omega-6, oleic acid omega-9) come from the oil itself, not from fish or animal fats.
The capsule shell, however, is described as a soft gelatin capsule. Gelatin is classically produced from animal connective tissue, and foods and supplements containing gelatin are not considered vegan by the usual definition. Some vegetarians also choose to avoid gelatin, while others accept it; it depends on how strict your personal dietary rules are. Because the manufacturer does not specify the gelatin source in the product information, the safest step is to contact them directly if this matters to you. If gelatin does not fit your diet, liquid flaxseed oil is an alternative that gives you the same plant oil without the capsule shell.
This is a food supplement, not a medicine, and it does not replace a varied diet.