I have a MacBook Air, model 6,2 with an Apple 500GB SSD. I want to upgrade to a new MacBook Air but I'd like to transfer the drive to increase storage on the new MacBook to 500GB.
Is this possible at an Apple store?
You can't "transfer the drive" as you describe from a 2013/14 MBA to a "new" (assuming 2018) MBA because the SSDs on the new unit are soldered onto the logic board.
To transfer your data, you have the following options available, ideally, using Migration Assistant
Can these things be done at an Apple store? The first two definitely can. It's highly unlikely they'll stock a USB enclosure. Also, keep in mind that since you can't upgrade the SSD on the new MBA, you must configure it with sufficient capacity (minimum 500GB) to fully accommodate your needs.
The new MacBook Airs have the T2 security chip and on-board SSD, which means you can't swap the internal storage without swapping the whole board.