This is somewhat redundant with Igor Pak's P.S., but I found an index for the 1918-1950 range in The Otto Dunkel memorial problem book, starting on page 80 of this pdf.
I found more: Stanley Rabinowitz and others had the same idea a couple of decades ago, and they started compiling indices, not just for the Monthly. Here are Google Books links for the two volumes I could find:
1975-1979 (published 1999)
1980-1984 (published 1992)