The organization responsible for evolving a shared vision of national development priorities, sectors, and strategies with the active involvement of States is the NITI Aayog.

"NITI" stands for National Institution for Transforming India. It was established on January 1, 2015, by the Government of India, replacing the erstwhile Planning Commission. NITI Aayog serves as a think tank and policy-making body for the Government of India, providing both directional and policy inputs.

Let's break down the answer into headings and key points:

1. NITI Aayog:
- NITI Aayog is a premier policy think tank of the Government of India.
- It was established to foster cooperative federalism by involving the states in the decision-making process.
- The organization acts as a catalyst of change and development, promoting the involvement of states in the policy formulation and implementation processes.

2. Evolution of a Shared Vision:
- NITI Aayog works towards evolving a shared vision of national development priorities, sectors, and strategies.
- It does so by engaging with the states and actively involving them in the decision-making process.
- The organization aims to create a platform for cooperative and collaborative federalism, where all the stakeholders work together to achieve the common goals of socio-economic development.

3. National Development Priorities, Sectors, and Strategies:
- NITI Aayog plays a crucial role in identifying and prioritizing national development goals and sectors.
- It formulates strategies and action plans to achieve these goals in consultation with the states.
- The organization focuses on key sectors such as healthcare, education, agriculture, infrastructure, and digital transformation, among others.

4. Active Involvement of States:
- NITI Aayog actively involves the states in the decision-making process.
- It organizes regular interactions, consultations, and meetings with the states to understand their specific needs and challenges.
- The organization encourages states to share their best practices and experiences, fostering a culture of knowledge exchange and learning.

In conclusion, the NITI Aayog is responsible for evolving a shared vision of national development priorities, sectors, and strategies with the active involvement of states. It acts as a platform for cooperative federalism, where the states and central government work together to achieve common development goals.