On the Atlas web interface all the values are shown correctly as Int64 values, but when I go to the same document using the plugin these values have changed to a double.

After some time has passed (or other unknown condition) the actual database values do change. Going from a visual bug to being what is actually saved. I save all my values as int objects and they do stay as int objects until it converts from this issue.

Is this an issue with the plugin or something else going wrong?

I believe I have now found the issue.
The VSCode plugin is displaying Int64 values as double, when you make a change to a document with Int64 values and save those values are set to double.
It does appear to be a bug in the plugin, I have reproduced this problem on a few different databases/collections

To Reproduce
Save a document that contains an Int64 value.
Open the same document via the plugin.
Make a change to the document and save.
Any Int64 values are converted to double.