Here are my basic directions for creating your own vegan yogurt, as well as places where you can purchase the culture.
You can easily make vegan/plant based yogurt out of soy or coconut milk (other plant based milks do not work well). However, they need to be pure without additional ingredients, stabilizers, sugar, salt, anything. WestSoy is a good brand that offers organic soymilk that is just soybeans and water. The only coconut milk I have found that is pure comes in a can. You want to get full fat coconut milk, and because you are buying a lot of cans it can get rather pricey. I have not personally tried making yogurt from coconut milk, but the process should be the same as with the soy.
Many vegan yogurt recipes call for you to boil the soymilk before you culture it. I personally find this time consuming, potentially messy (if you are not watching the pot it can boil over), and an unnecessary step. The soymilk already comes pasteurized and sealed in their containers. I have not had an issue so far using the soymilk straight from the box without boiling it. As with any packaged product, there is always a chance of damage to the container which may cause spoilage. I do a sniff test as a check before dumping the milk into the instapot. It's pretty obvious if the soy milk is spoiled, and if it is spoiled you wouldn't want to use it anyway (boiled or not).
You need something in which to culture the yogurt. We are conditioned to always expect our yogurt to come in neat little single serve containers. Yogurt can actually be made in a very large single batch, and either divvied out at the end into smaller containers, or consumed bit by bit from the original container. I personally like to keep the original container in the fridge, and just put a lid over it (I use a silicon stretch lid for kitchen food storage). To culture my vegan yogurt, I use an Instapot. It has a convenient yogurt setting that keeps it just the right temp. If you don't have or don't want to buy an Instapot, you can let the yogurt culture in an oven IF (and only if) you have a low enough temperature setting (108-112 F). There are other yogurt makers on the market, but I find the Instapot works great.
[FYI, to further counter your cultural conditioning about yogurt needing to come in single serve portions, in some countries, such as Greece, you can actually get yogurt portioned out for you in family sizes from a large barrel at the dairy deli counter.]
Basic Recipe
Put the desired amount of milk in the Instapot.
IMPORTANT: In a small bowl put some milk and put in the appropriate amount of your yogurt culture starter. Use a small whisk to get it mixed into a slurry.
Pour the slurry into the rest of the milk and stir well.
Set the time to culture your yogurt (8-30 hours).
Cover the pot with a light cloth.
Once finished, put in the fridge to chill.
I culture my yogurt for 30 hours to be as probiotic as possible (the longer the culture time, the more time the probiotic organisms have to reproduce). Note that the longer you culture the yogurt, the more sour it will be (which is a desirable). If I remember correctly, the minimum time for a good yogurt is 8-12 hrs. I like to cover the Instapot with a light cloth to keep things out, but not trap moisture in that might drip back down onto the yogurt.
After the yogurt has finished culturing, put it in the fridge to chill. Once cold, enjoy having great bountiful vegan yogurt! :o)
Below are some places to get cultures. Plant based yogurts always need a new starter culture with each batch, unlike dairy yogurts which can use a bit of the old batch to start a new batch. However, plant based is so much better than dairy, it is worth using a bit of new starter culture each time.
Cultures for Health: They have a truly vegan yogurt culture, as well as many other varieties of yogurt starter cultures. This company is a great place to get anything you need for fermentation. They also have water kefir grains to make your own water kefir.
Gi ProHealth: They offer GI ProStarter, which is a yogurt starter that I used for years to make soy yogurt very successfully. https://giprohealth.com/collections/gi-prostart%E2%84%A2-yogurt-starter/products/gi-prostart Visbiome: Note, this is not a vegan starter. Even though it is not vegan, I use it to create my soy yogurt because of GI health reasons. I was directed by my doctor to take Visbiome with lactulose to address SIBO, but taking it in that way can be very painful. I decided instead to use it to create yogurt. I do not know if all of the probiotic strains are cultured in the soy as they would be in cow, goat or sheep milk, but enough strains culture in the soy milk to definitely benefit my GI health.