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Abstract We report the discovery of three Milky Way satellite candidates: Carina IV, Phoenix III, and DELVE 7, in the third data release of the DECam Local Volume Exploration survey (DELVE). The candidate systems were identified by cross-matching results from two independent search algorithms. All three are extremely faint systems composed of old, metal-poor stellar populations (τ ≳ 10 Gyr, [Fe/H] ≲−1.4). Carina IV (M_V = −2.8;r_1/2 = 40 pc) and Phoenix III (M_V = −1.2;r_1/2 = 19 pc) have half-light radii that are consistent with the known population of dwarf galaxies, while DELVE 7 (M_V = 1.2;r_1/2 = 2 pc) is very compact and seems more likely to be a star cluster, though its nature remains ambiguous without spectroscopic follow-up. The Gaia proper motions of stars in Carina IV ( $M_{\star }=2250_{-830}^{+1180}{M}_{\odot }$) indicate that it is unlikely to be associated with the LMC, while DECam CaHK photometry confirms that its member stars are metal poor. Phoenix III ( $M_{\star }=520_{-290}^{+660}{M}_{\odot }$) is the faintest known satellite in the extreme outer stellar halo (D_GC > 100 kpc), while DELVE 7 ( $M_{\star }=60_{-40}^{+120}{M}_{\odot }$) is the faintest known satellite withD_GC > 20 kpc.