There is a way to use the inset tool like an extrude for a flat shape.
- Select the loop of vertices you want to extrude inwards.
- Fill in the area with a face F
- Inset the selected area I
- Delete the face
You do get small size variations in the new edge lengths around the curve compared to the straight sides but the amount is very small.
This only works gong inwards so you will want to scale your shape up to the larger size and then inset to the smaller size. With a bit of effort you can create a bounding box around your shape, fill it in with faces and press O while insetting to change to outset to enlarge your shape. Note in this example that the corners overlap and need cleaning.