I thought it kinda looked like jasmine, but we all know I'm better at dog training than plant ID.
Jasmine is not okay for them to eat. Does it have little white or pinkish flowers that smell like heaven?
Where are you located? That would help because jasmine is a zone 10-11 plant, maybe 8 & 9 with a good microclimate and care. Vinca grows zones 5-10 or so.
Look at these two pictures and compare your plant. Is it one of these?
The one on the left is Madagascar Jasmine.
The one on the right is one of the vincas.