You can construct the .vsix download URL by hand.

1. Get extension's "Unique Identifier", and split it into two parts along the .:

   eg. ms-python.python becomes ms-python and python

2. Get a version from "Version History" tab on marketplace.

   eg. 2024.17.2024100401

3. Determine the binary type that you need. Skip this step if this extension is "Universal". Note that some extensions do not support all binary types.

   Binary type	targetPlatform
   Alpine Linux 64 bit	alpine-x64
   Alpine Linux ARM64	alpine-arm64
   Linux ARM32	linux-armhf
   Linux ARM64	linux-arm64
   Linux x64	linux-x64
   Windows ARM	win32-arm64
   Windows x64	win32-x64
   macOS Apple Silicon	darwin-arm64
   macOS Intel	darwin-x64

4. Combine

   https://marketplace.visualstudio.com/_apis/public/gallery/publishers/ms-python/vsextensions/python/2024.17.2024100401/vspackage?targetPlatform=win32-x64

   The ?targetPlatform=win32-x64 part is optional, if your extension is universal.

In Python:

unique_identifier = 'ms-python.python'
version = '2024.17.2024100401'
target_platform = 'win32-x64'

publisher, package = unique_identifier.split('.')
url = (
    f'https://marketplace.visualstudio.com/_apis/public/gallery/publishers/{publisher}/vsextensions/{package}/{version}/vspackage'
    + (f'?targetPlatform={target_platform}' if target_platform else ''))
print(url)

You need to install it from a .vsix file. You can find them here.
Download the .vsix file of the version you want. You may have to click assets to see them.
Then open VSCode, go to extensions -> click on the three dots -> install from vsix and select your file.
To install the .vsix you can also use the command

code --install-extension ms-python-release.vsix

sources :
- How can I install Visual Studio Code extensions offline?
- https://code.visualstudio.com/docs/editor/extension-gallery#_install-from-a-vsix

#20184 explains how to download older versions of the extension.

Update [Since June 2024]:

This is the new template:
https://marketplace.visualstudio.com/_apis/public/gallery/publishers/ms-python/vsextensions/python/2024.15.2024091801/vspackage?targetPlatform=win32-x64

Valid targets: win32-x64, win32-arm64, linux-x64, linux-arm64, linux-armhf, darwin-x64, darwin-arm64, alpine-x64, alpine-arm64, web

Older version:

You can modify the URI template https://marketplace.visualstudio.com/_apis/public/gallery/publishers/ms-python/vsextensions/python/{version}/vspackage by replacing {version} with the version you want to download, e.g. https://marketplace.visualstudio.com/_apis/public/gallery/publishers/ms-python/vsextensions/python/2022.16.1/vspackage for 2022.16.1.

If you don't know which version you want and you have specific VS Code version restrictions, see the comment below on how figure out which version of the extension you want.