Homeopathic remedies act on a deep energetic level, and are potentized solutions (which can be then administered as liquid or pills) that are made by diluting then percussing or shaking, a wide variety of different substances to make each individual remedy. Usually a specific symptom pattern is matched to the symptoms the particular remedy would cause in a healthy individual. As far as vaccination goes Thuja is a remedy that is very often needed after vaccination and matches the symptom pattern of side effects (there are other remedies too but in this case it might be better to consult a homeopathic practitioner if other symptoms develop.) So giving the Thuja preventively can help prevent the symptoms ever developing, while still getting a healthy immunity. Some homeopaths do not do this, but I think it is worth while, and is deeper acting than Vit C but fine to use both together. Hope this makes sense, homeopathy is complicated! I am actually an animal homeopath (vet) but we had to learn about humans too, as that is where homeopathy was developed by Samuel Hahnemman.
PS I am sure it is better to split up the vaccines, at least it is in animals, and makes a lot more sense.