It is entirely possible to dye wool garments, but it's important to choose the right type of dye and method to prevent damaging the wool fibers during the process. To dye protein fibers such as wool, silk, fur, and feathers, acid dyes can be used. The dye is mixed into a dye bath with a low pH, and acid dyes are easy and safe to work with. The low pH is necessary for the dye to bind to the fibers, and the color should hold up well over time. We would recommend visiting a local hobby or yarn store where they can certainly help you with the choice of dye and provide you with more information on the process.
If you choose to dye your wool garment, it's useful to know that natural materials absorb dye, while synthetic materials will not do the same. Because the threads (seams) are made of 100% polyester, they will therefore remain in their original color. Before dyeing the garment, it's important to consider whether you'll be satisfied with the result even though the color of the seams will contrast with the rest of the garment.