$K$-shell x-ray production cross sections, $\frac{K\ensuremath{\beta}}{K\ensuremath{\alpha}}$ x-ray intensity ratios and $K\ensuremath{\alpha}$ and $K\ensuremath{\beta}$ x-ray energy shifts have been determined for thin solid targets of $_{15}\mathrm{P}$, $_{19}\mathrm{K}$, $_{20}\mathrm{Ca}$, $_{21}\mathrm{Sc}$, $_{22}\mathrm{Ti}$, $_{23}\mathrm{V}$, $_{25}\mathrm{Mn}$, $_{26}\mathrm{Fe}$, $_{27}\mathrm{Co}$, and $_{28}\mathrm{Ni}$ for 4-38-MeV $_{5}^{10}\mathrm{B}$ ion bombardment. Comparisons of the cross sections were made to direct Coulomb ionization and electron-capture theories by means of fluorescence yields corrected for multiple-ionization effects. The dominant contribution to $K$-vacancy production for these projectile-target combinations is believed to be direct ionization. Electron capture is expected to be important only for the lighter target elements and then primarily at the higher velocities. The direct-ionization theories employed were the binary-encounter approximation and the plane-wave Born approximation (PWBA) both of which overestimated the experimental data especially at the lower incident ion velocities. The PWBA was modified for increased target-electron binding, Coulomb deflection of the incident ion, polarization of the target-electron wave functions due to the passage of the incident ion, and relativistic target-electron velocities. The experimental data were found to agree quite well with the sum of the theoretical predictions of the modified PWBA and electron capture.