Block-based programming environments (BBPEs) such as Scratch and Code.org are now widely used in K-12 computer science classes, but they remain mostly inaccessible to blind or visually impaired (BVI) learners. A major problem is that prior accessibility solutions have relied on modifications to the Blockly library, making them difficult to apply in existing BBPEs and thereby limiting adoption. We present an Extension-based Accessibility Framework (EAF) to make BBPEs accessible for BVI students. The framework uses a modular architecture that enables seamless integration with existing Blockly-based BBPEs. We present an innovative three-dimensional (3D) hierarchical navigation model featuring stack labeling and block numbering, mode-based editing to prevent accidental modifications, and WAI-ARIA implementation to ensure compatibility with external screen readers. We evaluated our approach by integrating the EAF framework into two BBPEs (covering 177 test cases) and conducting semi-structured interviews with four participants using VoiceOver, JAWS, and NVDA. Participants reported clearer spatial orientation and easier mental model formation compared to default Blockly keyboard navigation. EAF shows that modular architecture can provide comprehensive accessibility while ensuring compatibility with existing BBPEs.