A VPN changes a different part of the connection
A VPN can be running while all of your usual browser extensions continue working exactly where they were.
You might notice the network change first. Reload ChatGPT after connecting to a VPN and the service may now see an IP address belonging to the VPN provider instead of the address from your home, office, or mobile connection.
Nothing unusual has happened to the browser itself, though. The same tab is open, the same page is being rendered, and the extensions you allowed on that site are still part of the browser.
That distinction matters because a VPN deals mainly with how traffic travels between you and the internet. It does not decide which Chrome extensions are allowed to interact with a page after it arrives in the browser.
So a writing assistant that was available inside ChatGPT before the VPN connected can still be available afterward. The same is true for another extension with permission to work on that site. Whether either one actually reads or sends chat content depends on what that extension was built to do.
VPN browser extensions make this a little less obvious because the same product can sit in both places.
Part of the extension may be responsible for sending browser traffic through the VPN service. That explains network or proxy-related permissions. The product may also have ordinary extension permissions for features that work directly with webpages.
Those are worth looking at separately.
If a VPN extension can work with the contents of an AI chat, the useful question is what feature needs that access and what happens to the information afterward. The fact that the product also provides a VPN does not answer either question.
Our guide to what a VPN actually protects covers the network side of this in more detail.