Certified GKH Gardening Expert
These appear to be Duchesnea rather than strawberry, though it is very easy to confuse them. the taste will tell you. Duchesnea is bland and features a yellow flower, while strawberries (even wild ones) will have a white flower, and be, either, sweet or sour.
Unfortunately, Duchesnea is much more invasive than strawberry. Also, very unfortunately, what kills those will kill your Vinca.
I would look into keeping seeds from your Vinca for planting next year, then kill off your entire bed. You can do this by removing all plants by hand, then tilling, then pouring boiling water over the entire bed a few times per day for a few days.
These methods, combined, will kill off EVERYTHING in that spot, while leaving the soil viable for planting your new seeds next year.
Adding landscaping fabric before you add soil and replant will help with the weeding, but it is always best to keep them handpicked, when possible.
This article will help you to amend your garden bed for next year's planting: https://www.gardeningknowhow.com/special/organic/the-importance-of-soil-for-an-organic-garden.htm