The launch of JWST has ushered in a new era of high precision infrared astronomy, allowing us to probe nearby white dwarfs for cold dust, exoplanets, and tidally heated exomoons. While previous searches for these exoplanets have successfully ruled out companions as small as 7-10 Jupiter masses, no instrument prior to JWST has been sensitive to the likely more common sub-Jovian mass planets around white dwarfs. In this paper, we present the first multi-band photometry (F560W, F770W, F1500W, F2100W) taken of WD 2149+021 with the Mid-Infrared Instrument (MIRI) on JWST. After a careful search for both resolved and unresolved planets, we do not identify any compelling candidates around WD 2149+021. Our analysis indicates that we are sensitive to companions as small as ~0.34 MJup outwards of 1."263 (28.3 au) and ~0.64 MJup at the innermost working angle (0."654, 14.7 au) with 5 sigma confidence, placing significant constraints on any undetected companions around this white dwarf. The results of these observations emphasize the exciting future of sub-Jovian planet detection limits by JWST, which can begin to constrain how often these planets survive their host stars evolution.