Vegan cheese has the lowest carbon footprint
First, let's take a look at cheese from dairy milk. Let's go with the lowest estimate and say that cheese has a carbon footprint of 5.13 kg CO2e/kg yield.
For cheddar, as sold at retail (63.2 % milk solids), the carbon footprint using the IPCC 2007 factors is 8.60 kg CO2e/kg cheese consumed with a 95 % confidence interval (CI) of 5.86–12.2 kg CO2e/kg. For mozzarella, as sold at retail (51.4 % milk solids), the carbon footprint is 7.28 kg CO2e/kg mozzarella consumed, with a 95 % CI of 5.13–9.89 kg CO2e/kg.
Source: Life cycle assessment of cheese and whey production in the USA
Now let's consider two different vegan cheeses with two different compositions. First up is Miyoko's cheese where the main ingredient cashews, has a carbon footprint around 2.3 kg CO2e/kg yield. This is likely to represent an upper bound because adding water (second ingredient) reduces the carbon footprint per kg of yield.
Miyoko's - Organic Cashews, Filtered Water, Organic Coconut Oil, Organic Parsley, Organic Rice Miso (Organic Rice, Water, Salt, Alcohol, Koji Culture), Organic Garlic, Organic Herbs, Sea Salt, Nutritional Yeast, Cultures.
And let's look at Daiya cheese. One life cycle analysis of cassava starch found that dry starch (like tapioca starch) had a footprint of 0.594 kg CO2e/kg yield. Again, this is likely to represent an upper bound because the first ingredient is just water. Vegetable oils (canola/safflower) can get as high as 4 kg CO2e/kg yield, but even if Daiya was made of 100% oil it would still be less than dairy cheese.
Daiya - Filtered water, tapioca starch, coconut oil, vegan natural flavours, pea protein isolate, non-GMO expeller pressed: canola and/or safflower oil, chicory root extract, sea salt, xanthan gum, lactic acid (vegan), tricalcium phosphate, pea starch, potato protein, vegan enzyme, cane sugar, annatto (colour), coconut cream