If you're using Bash, then the trigger that fires off this notification comes from the PROMPT_COMMAND variable in your shell. There is a different implementation for each shell. You can easily disable the notifications for one session by running unset PROMPT_COMMAND
in the embedded terminal after it launches. Unfortunately this turns off whatever else you have configured in that variable, it is frequently used for dynamic prompts (like a git prompt). If you can identify those parts of PROMPT_COMMAND that you want to keep, you can edit the variable to remove the elementaryOs additions.
To make this persistent in vscode, you can use the terminal.integrated.env.linux
preference to set PROMPT_COMMAND how you like it.
The elementary OS task finished notification specific part of the PROMPT_COMMAND is dbus-send --type=method_call --session --dest=io.elementary.terminal /io/elementary/terminal io.elementary.terminal.ProcessFinished string:$PANTHEON_TERMINAL_ID string:"$(history 1 | cut -c 8-)" int32:$__bp_last_ret_value >/dev/null 2>&1