Hello and thank you for your question. Even though Vinca is a popular plant it is considered aggressive.  After doing some research it seems cardboards or multiple layers of newspapers could help control the vinca growth. It would help to cut it back as far as possible before covering it.  You mentioned avoiding herbicides which is understandable. You could use a more eco friendly mixture of vinegar, salt and a squirt of Dawn dish soap on the primary growth plants if you can find them during cut back. This should be ok with the soil PH being on high side. I don't know how big the area is and would suggest working with a modest size area at first. You can cover with some mulch and water to keep paper/cardboard in place. Below are some links with additional information. I commend you for trying to have less lawn and  a more natural or productive yard. More people should be doing this!

periwinkle fact sheet

vinegar for weed control