We use observations with the infrared-optimized Magellan Adaptive Optics (MagAO) system and Clio camera in 3.9 ensuremathμm light to place stringent mass constraints on possible undetected companions to Sirius A. We suppress the light from Sirius A by imaging it through a grating vector-apodizing phase plate coronagraph with a 180° dark region (gvAPP-180). To remove residual starlight in postprocessing, we apply a time- domain principal-components-analysis-based algorithm we call PCA-Temporal, which uses eigen time series rather than eigenimages to subtract starlight. By casting the problem in terms of eigen time series, we reduce the computational cost of postprocessing the data, enabling the use of the fully sampled data set for improved contrast at small separations. We also discuss the impact of retaining fine temporal sampling of the data on final contrast limits. We achieve postprocessed contrast limits of 1.5 × 10$^-6$-9.8 × 10$^-6$ outside of 0.″75, which correspond to planet masses of 2.6-8.0 M $_J$. These are combined with values from the recent literature of high-contrast imaging observations of Sirius to synthesize an overall completeness fraction as a function of mass and separation. After synthesizing these recent studies and our results, the final completeness analysis rules out 99% of ensuremath≥9 M $_J$ planets from 2.5 to 7 au.