$ /bin/sh -c "$(cat foo); foo", the $(cat foo) part is evaluated in the outer sh process, so the actual argument your sh invocation is getting is:
$ /bin/sh -c "foo() {
printf "Hello World"
}; foo"
You have the function definition there and the call which works.
When calling
$ /bin/sh -c '$(cat foo); foo', the $(cat foo) part is evaluated in your sh process, so what's actually happening is $(cat foo) is trying to interpret the first "command" from the foo file, foo() which is obviously not found.