α Centauri A is the closest solar-type star to the Sun and offers an excellent opportunity to detect the thermal emission of a mature planet heated by its host star. The MIRI coronagraph on the James Webb Space Telescope can search the 1–3 au (1″–2″) region around α Cen A which is predicted to be stable within the α Cen AB system. We demonstrate that with reasonable performance of the telescope and instrument, a 20 hr program combining on-target and reference star observations at 15.5 μm could detect thermal emission from planets as small as ∼5 R ⊕. Multiple visits every 3–6 months would increase the geometrical completeness, provide astrometric confirmation of detected sources, and push the radius limit down to ∼3 R ⊕. An exozodiacal cloud only a few times brighter than our own should also be detectable, although a sufficiently bright cloud might obscure any planet present in the system. While current precision radial velocity (PRV) observations set a limit of 50–100 M ⊕ at 1–3 au for planets orbiting α Cen A, there is a broad range of exoplanet radii up to 10 R ⊕ consistent with these mass limits. A carefully planned observing sequence along with state-of-the-art post-processing analysis could reject the light from α Cen A at the level of ∼10−5 at 1″–2″ and minimize the influence of α Cen B located 7″–8″ away in the 2022–2023 timeframe. These space-based observations would complement on-going imaging experiments at shorter wavelengths as well as PRV and astrometric experiments to detect planets dynamically. Planetary demographics suggest that the likelihood of directly imaging a planet whose mass and orbit are consistent with present PRV limits is small, ∼5%, and possibly lower if the presence of a binary companion further reduces occurrence rates. However, at a distance of just 1.34 pc, α Cen A is our closest sibling star and certainly merits close scrutiny.