As a postdoc in the US, is it possible to apply for funding for a project, which can be used to support one or two undergrad/MS/PhD students?
The answer depends on the funding agency. Some agencies only require a PhD for the PI, in which case it's certainly possible, although it can be difficult to secure funding with only a postdoc as PI. (Many such proposals might have a more senior scientist as co-investigator.)
Other programs have specific requirements that the PI hold a more senior position, such as a tenure-track position, in which case a postdoc could not apply as the primary or sole or investigator.