Comets and asteroids preserve the least-altered materials remaining from the birth of the Solar System. However, the few visited by spacecraft are diverse in composition, and touring enough of these small bodies for a representative sample would be costly and time-consuming. A team of us is therefore developing a proposal for a mission to measure the makeup of large numbers of comets and asteroids via the dust they shed into the Zodiacal Cloud. The mission concept, Fragments from the Origins of the Solar System and our Interstellar Locale (FOSSIL), would carry four identical Dust Telescopes whose combined collecting area would yield measurements of the orbits and compositions of tens of thousands of the zodiacal particles, along with about a thousand of the interstellar particles that are constantly traversing the Solar System. These measurements would enable FOSSIL to reach four science objectives: (1) determine whether Jupiter-Family Comets' fine-grained components preserve unprocessed presolar molecular cloud particles, or show signatures of processing in the Solar System; (2) determine whether the rocky composition of the contemporary local interstellar cloud's dust population is consistent with being the feedstock for the formation of the Solar System; (3) determine whether comets' and asteroids' organics are genetically related, or formed from distinct reservoirs; and (4) determine the numbers and orbits of micron-sized particles originating from each of the three categories -- comets, asteroids, and interstellar dust -- in the Solar debris disk at 1 AU. FOSSIL's panoramic view of primitive Solar System bodies' chemical diversity, and its comprehensive measurement of interstellar dust sizes and compositions, would contribute to answering three of the major questions posed in the most recent Planetary Science Decadal Survey: What were the initial stages, conditions, and processes of Solar System formation, and the nature of the interstellar matter that was incorporated? What were the primordial sources of organic matter? And how have myriad chemical and physical processes shaped the Solar System? FOSSIL also would contribute to heliospheric science by measuring interstellar particles' deflection in the Solar wind, and to astrophysics through its measurements of the only debris disk where we can study individual particles and their parent bodies -- our own Zodiacal Cloud. Furthermore, by measuring both the particles' orbits and their compositions, FOSSIL would link decades of ground-based meteor trajectory data, lacking compositions, to decades of laboratory studies of meteorites and micrometeorites, lacking orbits. In particular, FOSSIL would tell us which families of bodies are the most likely sources of existing laboratory specimens. Each of FOSSIL's four Dust Telescopes consists of a Dust Trajectory Sensor (DTS) and an impact ionization reflectron-type time-of-flight mass spectrometer, the Composition Analyzer (CA). The DTS and CA are derived from and would use similar technology to previous and current flight instruments, especially the Cosmic Dust Analyzer (CDA) onboard Cassini, the Lunar Dust Experiment (LDEX) onboard the Lunar Atmosphere and Dust Environment Explorer (LADEE), and the Surface Dust Analyzer (SUDA) instrument in development for Europa Clipper. In summary, FOSSIL would be a low-risk mission to survey the composition of the Solar System's primitive material at a cost far lower than sending probes to a representative sample of small bodies. Building on in situ and returned sample measurements of cometary, asteroidal, and interstellar particles' compositions from the Giotto, Vega, Ulysses, Stardust, Rosetta, Cassini, and OSIRIS-REx missions, FOSSIL would enable breakthroughs in understanding our Solar System's largest visible structure, the Zodiacal Cloud, and reading its record of the origins of our planetary system.