Sure, you just need to be logged in.
Just to clarify - are you asking if your completion progress can be saved? If so, @JeremyLT is correct. Make sure you are logged in and you’ll see those satisfying little checkmarks.
But if you’re asking if your SOLUTIONS can be saved? I’m afraid FCC does not save solutions to lessons on their server. I believe that used to be a feature, but it was depreciated as the data took up too much space.