You don't the full fledged Salesforce DX here. To be able to use LWC, the following are required.

• Salesforce CLI. This is a must to be able to create projects, move components from sandbox to sandbox
• A code editor (preferably VS Code). You cannot use Developer Console to develop LWCs.

The following documentation provide more details on this topic:

• Set Up Your Development Environment
• Development Workflows

UPDATE: Updated the answer to reflect Metadata API support for LWC.

You don't need the full fledged Salesforce DX here. To be able to use LWC, you can use the following.

• Salesforce CLI. To be able to create projects, move components from sandbox to sandbox
• Metadata API. If not using CLI, you can use this to move components from sandbox to sandbox
• A code editor (preferably VS Code). You cannot use Developer Console to develop LWCs.

The following documentation provide more details on this topic:

• Set Up Your Development Environment
• Development Workflows
• LightningComponentBundle Metadata API