Near-Earth objects (NEOs) that orbit the Sun on or within Earth's orbit are tricky to detect for Earth-based observers due to their proximity to the Sun in the sky. These small bodies hold clues to the dynamical history of the inner solar system as well as the physical evolution of planetesimals in extreme environments. Populations in this region include the Atira and Vatira asteroids, as well as Venus and Earth co-orbital asteroids. Here we present a twilight search for these small bodies, conducted using the 1.2 m Oschin Schmidt and the Zwicky Transient Facility (ZTF) camera at Palomar Observatory. The ZTF twilight survey operates at solar elongations down to 35° with a limiting magnitude of r = 19.5. During a total of 40 evening sessions and 62 morning sessions conducted between 2018 November 15 and 2019 June 23, we detected six Atiras, including two new discoveries, 2019 AQ3 and 2019 LF6, but no Vatiras or Earth/Venus co-orbital asteroids. NEO population models show that these new discoveries are likely only the tip of the iceberg, with the bulk of the population yet to be found. The population models also suggest that we have only detected 5%-7% of the H < 20 Atira population over the seven month survey. Co-orbital asteroids are smaller in diameter and require deeper surveys. A systematic and efficient survey of the near-Sun region will require deeper searches and/or facilities that can operate at small solar elongations.