The policy of circular economy focuses on phasing out fossil-based packaging and replacing it with more sustainable alternatives. Companies face the challenge of choosing packaging for their products that are functional and affordable, and place relatively less pressure on the environment. This is especially important for organic farms that make voluntary commitments to undertake sustainable decisions regarding practices and methods of farming and types of packaging used. This publication attempts to analyze the determinants of the choices of sustainable packaging solutions made by organic farming companies with the example of Scilly Organic, an organic micro farm from the Isles of Scilly, United Kingdom—a producer of organic vegetables. There are many options for fresh vegetable packaging, which include fossil-based packaging, bio-based packaging, and packaging manufactured from material that is a mixture of synthetic, natural, or modified polymers. Biodegradable packaging, including compostable ones, is currently of particular interest because, when separated and disposed of in the correct manner in the waste management phase, they have sustainability potential. Biodegradable plastics constitute over 55.5% of global bioplastics production. Packaging is the largest market segment for bioplastic, with 48% of the total bioplastics market in 2021. Although the use of biobased packaging brings some advantages, it also comes with certain limitations that are the subject of intensive research. In this publication, the Life Cycle Assessment (LCA) tool was used and a critical review of the literature was carried out. Based on the analysis, the key factors and aspects influencing the environmental performance of selected types of packaging were identified. The LCA was carried out for the three selected packaging types, including low-density polyethylene (LDPE) bags, polylactic acid (PLA) bags, and polyester starch biopolymer (PCSB) bags. The research showed that the selection of more sustainable packaging is not straightforward. The analysis performed was the basis for providing recommendations for improving the sustainability of organic farms with regard to the selection of packaging for fresh vegetables. The critical processes in the life cycle that have to be considered are, in the first place, the production of polymer-based materials, and to a lesser extent, the production of the packaging bags and post-consumption waste utilization. In the case of PLA bags, 51% of the total impact is attributed to the production of polymer material. For starch polyester bags, this share is 58%, and for LDPE it constitutes 41% of the total score. At the same time, the choice of packaging should be made in the context of the specific properties of the packaging material, the requirements for disposal methods, and local waste management systems.