Safe to charge swollen 2015 MacBook Air battery?

No, it is absolutely not safe to attempt to charge or us a swollen battery.

Also, please keep in mind that it is illegal to "send" a damaged Lithium-ion battery by pretty much ANY means. You can't "send" it anywhere.

You can drop it off at your local Apple Store or electronics recycler or if your community has a hazardous materials drop off, you can potentially drop it off there. You can not legally mail or ship it.

Since you plan to recycle the computer, You might consider removing the SSD from the device and install it into an external enclosure to access the data that is on it and perhaps even continue to use the SSD as an external drive with another computer.

You can find guidance online for this task at macsales.com. Depending on the exact year and model MBA, the removal of the drive is probably very straightforward. Macsales also can provide an enclosure to house the SSD.

Nobody here can guess if your swollen battery will catch fire in the few minutes it takes to get your photos. It is advisable not to charge a swollen battery due to this fire risk.

Seriously ??? 🫣

Of course it’s NOT ok.

Is this the 13" or 15" model? Apple has a free repair program to address a potential fire safety issue (Battery Recall) for the 15" model which apparently is still active (it is still listed on Apple's site for free repair programs):

15-inch MacBook Pro Battery Recall Program - Apple Support

If you have the 15" model and your laptop's serial number is confirmed to be part of the Battery Recall program, then you may not want to use that battery. If your laptop is not part of that Battery Recall, then you should be Ok to quickly pull your data from the laptop. Since I know the others object...to be extremely careful, you can take the laptop outside & use an external keyboard & mouse to keep the laptop away from anything flammable as well as to allow you to keep a bit of distance. Have a small container of sand handy ready to throw onto the laptop just in case of a fire so it can be safely smothered.

Apple doesn't consider a swollen battery as a fire hazard. In fact Apple only ever had one free repair program which was to address a fire safety hazard (the one I linked above). Apple had other free repair programs to address swollen batteries. Apple never mentioned any fire safety issues or any real safety issue with the swollen batteries programs. Of course you never want to have a battery leak any of the chemicals or in the case of a swollen battery... any of the gas since both are toxic.

While you usually don't want to use the laptop with a swollen battery, in your specific case where you only need to use it for a very short while to retrieve data.....you should be Ok to do so. You may find that the battery will increase its swelling which could cause the Bottom Case to warp and to push out the Trackpad. You should definitely have an external keyboard & mouse handy.

FYI, in the future you should make sure to always have frequent & regular backups because there are a lot more new ways to permanently lose access to the data stored on the internal SSD of the recent Macs due to all of the hardware, software, and security changes.

You're welcome.