Ambiguous language and confusion over official messaging has driven this claim to spread so far and wide that many pro-vax even believe it. It is of course true that the vaccine is not 100% effective at preventing infection. It isn't 100% effective at anything... no vaccine is. But they are VERY effective at preventing infection.
Based on US county level case surveillance and vaccine administration data, "A 10% improvement in vaccination coverage was associated with an 8% (95% confidence interval 8% to 9%) reduction in mortality rates and a 7% (6% to 8%) reduction in incidence.
"Our cohort study in healthy workers conducted from the end of the first wave confirmed that reinfection after natural infection is seven times more likely than infection after vaccination. This finding supports the CDC recommendation that all eligible persons be offered COVID-19 vaccination, regardless of previous SARS-CoV-2 infection status."
A study where 3975 health care personnel were tested weekly for 5 months...
"SARS-CoV-2 was detected in 204 participants... Adjusted vaccine effectiveness was 91% (95% confidence interval [CI], 76 to 97) with full vaccination and 81% (95% CI, 64 to 90) with partial vaccination."
"Vaccination reduces the risk of delta variant infection and accelerates viral clearance."
"Vaccine efficacy against all infections (asymptomatic, symptomatic, ambulatory or hospitalized) caused by all SARS-CoV-2 is greater than or equal to 89% in people who have received two doses of mRNA vaccines" (translated)
During a Delta variant outbreak in a prison with frequent testing...
"The estimated vaccine effectiveness was 56.6% against infection and 84.2% against symptomatic infection"
"VE exceeded 90% against SARS-CoV-2 infection when at least one dose was an mRNA vaccine... Two-dose mRNA VE was maintained against hospitalization for the 5-7-month monitoring period and while showing some decline against infection, remained ≥80%."
"But most cases are vaccinated"
In populations with a high vaccination rate, it is often the case that the majority of cases are in vaccinated individuals, while the case rate is still lower in vaccinated. Forming an argument based on the raw numbers while the rate is actually lower is what is known as the "base rate fallacy".
"But the case rate is higher in vaccinated"
In late December 2021, some regions are seeing higher case positivity in vaccinated individuals. This may be because vaccinated individuals are more likely to get tested (we await further data on this). The rising case positivity in vaccinated individuals is probably also driven by reduced vaccine effectiveness vs infection for Omicron variant, as well as waning immunity vs infection, as a large proportion of the population is 6 or more months from second dose and not yet boosted.