закваска which is an ingredient in Bulgarian yogurt is translated as
starter in English. Is this ingredient vegetarian?
Although you mention Google translating
закваска ('zakvaska' in Latin alphabet) as yeast, I believe it means probiotic cultures instead.
This does not change much in respect of it being vegetarian, since both yeast and probiotic cultures are vegetarian. Although probiotic cultures are sometimes called live cultures, due to their simplicity and omnipresence, they are not considered as problematic for vegetarians, nor vegans (that is not to say that yogurt is vegan, it is not, but for different reasons).
The essential part of the starter is a bacteria culture (some kind of lactobacillus). Yogurt is always made with a bacterial culture.
Bacteria can be eaten by vegetarians and vegans: they are not animals. They are among the simplest life forms, lacking even a cell nucleus (they are prokaryotes). They are far less complex than plants. Countless numbers of them are harboured and slain by our bodies without any intention on our part.
However, yogurt may sometimes contain gelatin or other non-vegetarian additives.
From what I read regarding yoghurt production, the main (starter) cultures in yogurt are Lactobacillus bulgaricus and Streptococcus thermophilus. The starter cultures are used to convert milk to yogurt during fermentation. As others have mentioned, these are friendly bacteria and not an animal derivative. If you are a strict vegetarian like me, then what we need to look out for mainly is gelatin as Zanna mentioned. Whenever I am not sure what the ingredients are or when the ingredient label is not clear, I always call or email the company to clarify. You may wish to do the same too.
Yoghurt is probably not vegan if you buy it from the store because they all contain sugar which we know is filtered through bone char made from animal bones. Definitely other ingredients makes it non vegan or non vegetarian. Other thing which makes it non vegan is a milk itself which here in usa cows are suffering a lot by producing milk. Women has milk when she delivers baby so would be same for a cow but instead they are injected with hormones to have milk all the time regardless of having a calf or no, which causes lot of pain and suffering for cow. In India in many cases cow is considered as sacred animal and it is treated well and in many cases not killed for food. In many places in India it is illegal to kill the cow. Among Hindus cows are symbol of mother and prosperity and they worship cows and treat them with respect. Those cows die their natural death. Milk from such cows is taken with great care by not harming a calf in any way. Such milk is fine to drink for vegans or vegetarians because there is no harm involved but pure love for cow which is treated as a god or goddess .It is definitely deeper science involved than we know. Forgot to add also as a starter you can use lemon for making yoghurt or cheese as well but not sure of starter used for store brought yoghurt it is probably not vegan