First scale Rate to Rate*max(df$Response) and scale the text 0.9 of the response text.
Second, enable the second axis via scale_y_continuous(sec.axis=...) :
ggplot(df) + geom_bar(aes(x=Year, y=Response),stat="identity", fill="tan1", colour="sienna3")+ geom_line(aes(x=Year, y=Rate*max(df$Response)),stat="identity")+ geom_text(aes(label=Rate, x=Year, y=Rate*max(df$Response)), colour="black")+ geom_text(aes(label=Response, x=Year, y=0.95*Response), colour="black")+ scale_y_continuous(sec.axis = sec_axis(~./max(df$Response)))
What gives:

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