If you suffer from anxiety, talk to your doctor about anxiolytic medication, at least for the flights. A plane is a very enclosed space, and some people suffer from adverse effects when exposed to vape fumes. Not only that, others find it off-putting and would rather not be forced to breathe that in during a flight.
As others have correctly pointed out, KLM does not allow vaping on board. If you do so, you can cause a flight to be diverted, and even have a suit brought against you for damages caused by your disruption to the flight.
Using an electronic cigarette or recharging it or its batteries is never allowed on board.
In terms of legality, the FAA explicitly says that vaping on board is a federal offense. So if you do this in US airspace, expect cops upon landing.
Just as passengers are not allowed to smoke cigarettes on an aircraft, they should never use their vapes or e-cigarettes on an aircraft. It’s not only dangerous, it’s a federal offense.