Unless your law is very different from the ones I have seen, YOU have to proceed with the lemon law, not the car company. But if the company agrees to repurchase the car, then the company moves forward with it.
I went through this process in Arizona a number of years ago on a BMW. I went to the dealership with a letter stating that I was initiating the lemon law process. It helps that I am a retired litigator. The service manager at the dealership told me he agreed with me and asked me to give him an hour. An hour later he told me that BMW had agreed to buy back the car. So I went through that process instead of the lemon law process. With BMW, BTW, it was painless. They made me a fair offer to account for my usage and even let me keep the car while I waited for a replacement BMW to be built to my requests.