I make lactose free yogurt all the time with various probiotics and have also used yogurt starters. Most bacteria are able to utilize more than one type of sugar and not just lactose. There are also commercial soy yogurts. I have also tried with almond milk and coconut milk and one of those did not work- I forget which one. Just don't expect thick and creamy store-bought kids of yogurt- many of those add starch to thicken them up. I use sterile jars and depending on time of year, either stick them outside ( I live in the desert so temps are 90-100 for several months of the year- but not in direct sun) or indoors in a water bath.
I then use these home-made probiotic yogurt to make smoothies. I will say that when I have to travel for work and am unable to have my home-made smoothie, I do suffer more, but that could also be just the stress of traveling on the road, staying in hotels, eating crap food when the client takes you to lunch, etc