Vega has been shown to host multiple dust populations, including both hot exo-zodiacal dust at sub-AU radii and a cold debris disk extending beyond 100 AU. We use dynamical simulations to show how Vega's hot dust can be created by long-range gravitational scattering of planetesimals from its cold outer regions. Planetesimals are scattered progressively inward by a system of 5-7 planets from 30-60 AU to very close-in. In successful simulations the outermost planets are typically Neptune-mass. The back-reaction of planetesimal scattering causes these planets to migrate outward and continually interact with fresh planetesimals, replenishing the source of scattered bodies. The most favorable cases for producing Vega's exo-zodi have negative radial mass gradients, with sub-Saturn- to Jupiter-mass inner planets at 5-10 AU and outer planets of 2.5 to 20 Earth masses. The mechanism fails if a Jupiter-sized planet exists beyond ~15 AU because the planet preferentially ejects planetesimals before they can reach the inner system. Direct-imaging planet searches can therefore directly test this mechanism. ; Comment: Updated references. Accepted to MNRAS Letters. 5 pages, 4 figures. Blog post about the paper at http://planetplanet.net/2014/03/31/vega-a-planetary-poem/