The origin of ferroelectric polarization in tetragonal tungsten-bronze- (TTB-) type oxide strontium barium niobate (SBN) is investigated using first-principles density functional calculations. We study in particular the relationship between the polarization and the cation and vacancy ordering on alkali-earth metal lattice sites. Lattice dynamical calculations for paraelectric structures demonstrate that all cation configurations that can be accommodated in a $1\ifmmode\times\else\texttimes\fi{}1\ifmmode\times\else\texttimes\fi{}2$ supercell result in a single unstable polar phonon, composed primarily of relative Nb-O displacements along the polar axis, as their dominant instability. The majority of the configurations also have a second octahedral tilt-mode instability which couples weakly to the polar mode. The existence of the tilt mode is strongly dependent on the local cation ordering, consistent with the fact that it is not found experimentally. Our results suggest that ferroelectricity in the SBN system is driven by a conventional second-order Jahn-Teller mechanism caused by the ${d}^{0}$ ${\mathrm{Nb}}^{5+}$ cations, and demonstrate the strong influence of the size of Sr and Ba on the lattice distortions associated with polarization and octahedral tilting. Finally, we suggest a mechanism for the relaxor behavior in Sr-rich SBN based on Sr displacement inside pentagonal channels in the TTB structure.