Can I solder more memory into my MacBook Air?
iFixit, who is a proponent of replacing everything in your system, does not provide a guide to replace the RAM (only the entire logic board, on which the RAM is soldered). This alone should tell you that it is something that should not be attempted by users who are not professionals.
One user did ask in their forums if it was possible. Here's one response:

...to replace/upgrade/or modify those chips, we send them to factory. Precision soldering is required, it is not to be done at home (too fine a work for home, unless you have done this same exact work before and done it well everytime).
Even after factory re-work, some boards/assembly simply fail (shorted etc). Definitely not worth the risk unless you have direct access to a mother board assembly factory and someone does it for you over there.

Another user said:

In any case SMT chips as densely laid out as these are can't be desoldered and new ones desoldered by hand. You need very special gear only available at a fab house.

And a third user added:

They use a special fixture to hold the chips in place and heat both sides at the same time. Your chances is less than 20% getting it to work and heating the board without the proper masks could cause the other components to shift killing your system for sure...

Given that you can sell your (presumably functioning) existing machine for a fair amount of money, it seems like it would be much more practical to buy a new machine than take a very high chance of destroying your current one, since then it will be worth next to nothing.