First off, not much is known about the impact of the lead vaccine candidates in terms of reducing transmission. See https://www.cbsnews.com/news/covid-19-vaccine-infectious/, which mentions that we simply don't have the clinical trials to draw conclusions yet. It is possible that vaccines will only grant the vaccinated immunity but have no impact on it continuing to spread. But assuming that isn't the case...

The amount of immunity needed to achieve herd immunity (where spreading is so rare that the virus ceases to circulate) depends on the spreadability of the virus. This is often represented by R0, but that figure can be skewed by various factors and in turn skews herd immunity targets. For something very infection like measles, you need immunity rates above 90% to achieve herd immunity. For SARS-CoV-2, given the wide range of estimates on R0, the ranges of immunity needed are estimated to be from 10% to 70% (see https://www.medrxiv.org/content/10.1101/2020.07.23.20160762v... and https://www.medrxiv.org/content/10.1101/2020.04.27.20081893v...), with consensus being that we need above 60%. We have some level of immunity in the population already from those who have been naturally infected and have recovered - for example a recent December update on Houston's antibody survey suggests 13.5% of their population previously had COVID-19 (http://www.houstontx.gov/health/NewsReleases/houstons-covid-...). We'd need to use either continued natural spreading or interventions like vaccination to increase the percentage of population that is immune.

The closest I've found to what you're asking for is at https://www.statnews.com/2020/12/17/calculating-our-way-to-h..., which shows different timelines to achieving herd immunity based on the percentage of the population that is vaccinated. If only 39% get vaccines, assuming those who are vaccinated do not spread the virus, then we would achieve herd immunity by December 2022 (because natural infections would take us the rest of the way to the herd immunity threshold). However if we just directly vaccinate to the necessary threshold, then we'd achieve herd immunity as soon as those vaccinations are performed.

Below the herd immunity threshold, different levels of vaccination also reduce the amount of fatalities drastically depending on the distribution strategy. COVID-19 has very low fatality rates for those under 50 (https://www.nbc26.com/news/coronavirus/cdc-estimates-covid-1...) and so if you vaccinate the elderly, healthcare workers, and the immune compromised, then the dangers of the virus substantially disappear. As an example, if everyone above 50 in the US were vaccinated and everyone under 50 were infected, we would expect a maximum of around 30,000 deaths from the under 50 population, based on the infection fatality rates. And of course, the rate of spreading being slower from increased immunity (even if below the herd immunity threshold) means that more medical resources are available due to the flattened curve.

The other question is whether we will see antigenic drift that requires continual vaccine updates like with a flu vaccine. Emerging evidence suggests this is what we should expect - see this thread (https://threadreaderapp.com/thread/1340409968818671616.html) from Trevor Bedford (https://www.seattletimes.com/seattle-news/health/a-conversat...). If there is antigenic drift that we aren't able to "keep up" with from vaccination, then SARS-CoV-2 will likely just be endemic (meaning there will always be some background rate of humans who are infected).