https://www.fda.gov/drugs/development-approval-process-drugs provides some general information on how the FDA makes their decisions. There is no general "efficacy" threshold that must be met for approval.
Summarizing the bullets on that site, approval depends on:
Analysis of the target condition and available treatments
If there are already treatments for something, FDA will consider the efficacy of those treatments in approval. If the target condition is something cosmetic, they aren't going to tolerate much as far as side effects; if it's something life-threatening, then serious side effects may be acceptable if it saves lives on net. Vaccines in particular must be very safe for broad approval, because many of the people getting the vaccine don't currently have the illness.
Assessment of benefits and risks from clinical data
Trials give information about both safety and efficacy. Both will be considered along with considering the disease characteristics and available treatments.
Strategies for managing risks
This is pretty much the labeling, and deciding on any restrictions (for example, not giving a novel drug to pregnant women if it wasn't tested in pregnant women; for a hypothetical HIV vaccine perhaps to target vulnerable groups only rather than the general population) to limit possible concerns.
The FDA will make decisions about vaccine approval based on the seriousness and prevalence of the disease, safety and efficacy of a given candidate vaccine, and presence of other approved competing vaccines.