The introduction of alternative crops, including wild edible and medicinal plants, in organic cultivation systems presents an attractive approach to producing healthy and high-quality products due to their content in beneficial compounds and increased nutritional value. The current study evaluated the impact of organic and conventional fertilization on the growth, quality, nutrient status and stress response of the two wild edible species, e.g., purslane (Portulaca oleracea L.) and common sowthistle (Sonchus oleraceus L.), under field conditions. The fertilization treatments included the following: a control (NoFert) treatment with no fertilizers added, base dressing with conventional fertilization (CoFert), base dressing with organic fertilization (OrFert), base dressing and side dressing with conventional fertilization (OrFert + SCoFert) and base dressing and side dressing with organic fertilization (CoFert + SCoFert). Organic fertilization was carried out using a commercial vinasse-based organic fertilizer. In both purslane and common sowthistle, the application of organic fertilizers provided comparable or even enhanced plant growth traits, macronutrient content (i.e., P and K for purslane, and N for sowthistle) and quality (i.e., total soluble solids) compared to the application of conventional fertilizers. On the other hand, conventional fertilization with supplementary fertilization positively influenced the plant growth of purslane (i.e., plant height and stems biomass), as well as its physiological parameters (i.e., chlorophylls content), total phenolics content and antioxidant capacity (i.e., DPPH and FRAP). Similarly, conventional fertilization led to increased total phenolics and antioxidants in common sowthistle, while variable effects were observed regarding plant physiology, stress response and antioxidant capacity indices. In conclusion, the use of organic fertilization in both purslane and common sowthistle exhibited a performance similar to that of conventional fertilization, although further optimization of fertilization regimes is needed to improve the quality of the edible products.