The question as asked cannot be answered, because the number of people who avoid death because of a vaccine depends on the size of the population (and on the general material level of life and hygiene of the population), and the size of the population itself in a world without vaccines is an unknown which cannot be estimated reliably.
We can however make a reasonable estimate of how many people were killed by a specific disease in a specific territory in a specific time frame. For this exercise I chose to estimate how many people were killed by smallpox in Europe from 1750 to 1850.
To compute this sum we need (1) the ratio of deaths from smallpox over the size of the population, and (2) the actual size of the population.
In Europe, as of the end of the 18th century, an estimated 400'000 persons died annually from smallpox, and survivors accounted for one third of all cases of blindness.
(From Donald A. Henderson and Bernard Moss, "Smallpox and vaccinia", in Stanley A Plotkin, MD and Walter A Orenstein, MD, Vaccines, 3rd edition, Philadelphia, 1999.)
Henderson and Moss don't say what exactly they refer to when they say "end of 18th century Europe", but let's take the population of 152 million given by Encyclopedia.com for "Europe without Russia", whatever that means. The same source gives a population of 125 million for Europe without Russia in 1750. Assuming constant morbidity and mortality, that gives about 18.2 million people killed by smallpox alone in Europe in the last 50 years of the 18th century; during the 19th century the population of Europe grew at faster pace, reaching 208 million in 1850. With the same assumptions we get 23.1 million people killed by smallpox in Europe in the first 50 years of the 19th century.
In total we have some 40 million people killed by smallpox in Europe alone in the 100 years between 1751 and 1850. For comparison, the deadly Napoleonic wars killed about 3.1 million people (worldwide, but mostly in Europe) from 1803 to 1815.
I have no idea how to extrapolate those numbers to reach the present, because after 1850 smallpox vaccination became more and more widespread, with almost universal coverage in western Europe since about 1920 or so, and in all of Europe since about 1950 or so.
Before the introduction of universal vaccination child mortality was very high; for example, in Europe from 1800 to 1850 about 1 child in 4 died before the age of 5 (from Our World in Data), and 1 in 2 died before the age of 15. This is much too much different from the modern world; just for example, in a world were each woman needs to give birth to 4 or 5 children just to keep the population stable, women will tend to have a more limited participation in the workforce, with very large effects on the economic growth.