By the looks of it, I believe so. They have a webpage for all the types of downloads here. Simply replace {version} with the version you want, or latest for the latest version.

After digging through the official site, I discovered that a list of releases exists on the Visual Studio Code Updates page and each change list entry there includes links to the associated downloads.

Not every version is listed, but there is (at this time) over a year's worth of releases displayed there.

Click on a month's release notes and you'll find download links for that version at the very top of the page just under the headline. Note that not all downloaded files include the version number, so you may want to rename them to include it as you download.

Edit: It appears that all of the old change lists still exist even if they are not on the left-hand navigation. The trick is that you have to know the major and minor version number. These pages link to the last patch versions for the requested version, so you don't have to add the patch version to the URL.

https://code.visualstudio.com/updates/v<MAJOR>_<MINOR>

For example:

https://code.visualstudio.com/updates/v1_20 (1.20.1)
https://code.visualstudio.com/updates/v1_35 (1.35.1) (Last OS X Mavericks version)
https://code.visualstudio.com/updates/v1_40 (1.40.2)
https://code.visualstudio.com/updates/v1_54 (1.54.3) (Last OS X Yosemite verison)

If you need an older patch version than the one linked from those pages, you can just edit the download link. For example, to retrieve 1.54.1, go to the v1_54 page and change the download link from "/1.54.3/" to "/1.54.1/"

If you're looking for version numbers or to get the source for a particular version, your best bet is probably to browse the release/tags list on the Visual Studio Gode GitHub repository.

Edit: Thanks to VSB's comment, it appears that Microsoft has put up a FAQ detailing how to navigate to direct download links of old versions for all platforms.

It follows the same basic idea as my original answer above but with a slightly different structure to indicate the platform and installer type (e.g. win32-x64, win32-x64-user, win32-x64-archive) and build (stable, latest):

https://update.code.visualstudio.com/<version>/<platform>/<build>

So a 64bit Windows, system installer, stable build for v1.54.1 would be:

https://update.code.visualstudio.com/1.54.1/win32-x64/stable

The list of platforms shown is as follows:

• win32-x64
• win32-x64-user
• win32-x64-archive
• cli-win32-x64
• win32-arm64
• win32-arm64-user
• win32-arm64-archive
• cli-win32-arm64
• win32
• win32-user
• win32-archive
• cli-win32-ia32
• darwin-universal
• darwin
• cli-darwin-x64
• darwin-arm64
• cli-darwin-arm64
• linux-x64
• linux-deb-x64
• linux-rpm-x64
• linux-snap-x64
• cli-linux-x64
• linux-armhf
• linux-deb-armhf
• linux-rpm-armhf
• cli-linux-armhf
• linux-arm64
• linux-deb-arm64
• linux-rpm-arm64
• cli-linux-arm64

It's possible now, Microsoft released Linux 32bit version