Vegan thousand island dressing (Loving It Vegan) is made with vegan mayo, ketchup, garlic and onion powder, paprika and distilled white vinegar (the edible not cleaning kind, sold in most groceries). Also try this recipe for vegan blue cheese dressing made with cheesy-tasting nutritional yeast and white miso (avoid unpasteurised for pregnancy/nursing)
Read up on food safety for people & pets (many garlic, onion and citrus are unsafe near animal friends). Just bin allium scraps (onion, garlic, leeks, scallions, chives) as like citrus and rhubarb scraps, acids could harm compost creatures.
The Science of Salad Dressings
To make a salad dressing, you simply mix an acid (like lemon juice or vinegar) with an emulsifier (say mustard). Then swiftly whisk in your oil, so it breaks into tiny droplets, to mix in with the other ingredients. Creamy dressings are good made with vegan yoghurt or mayo.