Chicago Manual of Style features many examples of footnotes without spaces, and says nothing to contradict that practice. Also, from over a decade of experience with Chicago, I understand footnotes should follow directly after what they note without a space.
All of these examples of usage are in 14.26: Placement of note number in the online version of the Chicago Manual of Style, 17th edition. I have preserved the spacing exactly, only adding some superscript code (
<sup></sup>) so it appears correctly in StackExchange:
“This,” wrote George Templeton Strong, “is what our tailors can do.”1
It was the hour of “national paths” toward socialism;9 but that
expression, which turned out to be temporary, was more an incantation
than a discovery.
The bias was apparent in the Shotwell series3—and it must be
remembered that Shotwell was a student of Robinson’s.
(In an earlier book he had said quite the opposite.)2
Men and their unions, as they entered industrial work, negotiated two
things: young women would be laid off once they married (the commonly
acknowledged “marriage bar”1), and men would be paid a “family wage.”