I just adopted a puppy who is around 8 months old. The rescue organization that I got her from told me she contracted distemper when she was around 3-4 months old, just after her first vaccine. The woman who I adopted her from was hesitant to give her the last two shots of the distemper vaccine because she got so sick from the actual virus that she almost died. That being said, if there is a non-live virus vaccine for distemper that I can give her I think this would be my best option forward. If anyone knows of such a vaccine please let me know.
I recommend that you talk to your vet about your concerns. The UC Davis Veterinary Medicine website says
Dogs that recover from natural infection have lifelong immunity to CDV.
That suggests to me (but again, talk to your vet) that your puppy is in no danger from a live vaccine. If there is a non-live vaccine, your vet should know.
I have no medical training, but my layman's guess is that the puppy did not contract distemper from the vaccine; she was just exposed to it around the same time (either before the vaccine or before it could build up her immunity). "Live" vaccines normally contain a weakened form of the virus that should not cause the disease.
I understand your concern. Even after talking to your vet, it might be that your brain is convinced that the vaccine will be safe for your puppy, but your heart is not convinced. If that's the case, and if there is no killed vaccine for distemper, your vet might have other suggestions. A good vet will be sympathetic to your concerns.
Update: The wording on this page seems to suggest that there is a "killed recombinant CDV vaccine" (CDV = Canine Distemper Virus).