The GSS suggests it occurs. So does the Cooperative Congressional Election Study of 2016, which surveyed 38,465 people who reported voting in a presidential primary during the last election cycle. Of those, 259 were classified as non-citizen immigrants. That comes to 0.7% of the active primary electorate. The party distribution of their primary/caucus party participation:
This isn’t enough to materially affect the outcomes of the vast majority of state elections. Assuming a similar general election preference distribution, using this data we can reasonably estimate that non-citizen voting potentially nets the Democrat candidate about 0.5 points over the Republican candidate. In 2016, the only state that would’ve flipped was New Hampshire. Hillary Clinton won the state by 0.37% of the vote. New Hampshire is less foreign-born than the country as a whole, though, so it may not have even amounted to enough to change the outcome in the Free State.
On the other hand, it’s not nothing, and any individual citizen has the right to be upset by the idea that someone who shouldn’t be casting a ballot could be cancelling out his constitutionally legitimate vote.
Parenthetically, American Samoans are legally American nationals. They are not citizens like the native-born residents of the other American territories are. The population of American Samoa comprises less than 0.002% of the empire’s total, however, so unless they are oversampled by a factor of 350 here, they don’t account for the non-citizen vote!
CCES variables used: IMMSTAT, CC16_328