Jacob Kaplan's book A Criminologist's Guide to R has a section on downloading different file types. As for .txt files in ASCII (which my file type was), he writes "hopefully you'll never encounter" this "very old file format system" (p. 51). To read it into R, he created the asciiSetupReader package. I installed that package and used it on my ASCII .txt file. It still didn't work. So I downloaded the NIBRS file in SPSS format and tried read_sav() (from the haven package). This still didn't work. So I used the asciiSetupReader function read_ascii_setup() with the SPSS file:

NIBRS2014_1 <- read_ascii_setup("C:/Users/steve/OneDrive/Desktop/RaceEconProfiling/NIBRS/ICPSR_36421_SPSS_2014/DS0002/36421-0002-Data.txt" , "C:/Users/steve/OneDrive/Desktop/RaceEconProfiling/NIBRS/ICPSR_36421_SPSS_2014/DS0002/36421-0002-Setup.sps")

This worked!