There are two answers here.
NO: There is no need to split up vaccinations over multiple visits for most animals. Most dogs and cats do fine with multiple vaccinations at once. It is also inconvenient to make multiple appointments. However…
YES: Reactions do happen. They are rare, but we see a reaction about once a year. If multiple vaccines were administered, we have no way to know which caused the reaction. Separating the vaccines over multiple visits gives us better data… we don’t want to just not ever vaccinate again, but we do want to know which vaccines we need to rethink, and if still needed we can take extra precautions in the future.
Most pet owners are very busy people and prefer to make just one trip to the vet. This is a very good question for you to ask YOUR vet at your dog’s/cat’s next wellness visit.