jmc

Note that $$\begin{array}{rl}f(x)+f(-x)& =(a{x}^7+b{x}^3+cx-5)+(a(-x{)}^7+b(-x{)}^3+c(-x)-5)\\ & =(a{x}^7+b{x}^3+cx-5)+(-a{x}^7-b{x}^3-cx-5)\\ & =-10.\end{array}$$In particular, $f(7)+f(-7)=-10,$ so $f(7)=-10-f(-7)=\boxed{-17}.$