I have a Macbook Air 2014 - 128 GB, and I am running out of space. Is it possible to upgrade the SSD to one of bigger capacity?
In some Macbook Airs the SSD is actually replaceable. Please refer to iFixit.com and the other answers for details.
However, a very interesting option is to make use of the SD card slot. There are a number of products out there that will fit seamlessly into your SD card reader slot. You can then mount it as an additional drive and put e.g. your music library onto it. Some examples (feel free to update):
Using one of these solutions you could add up to 256 GB (2015) of storage to your Mac.
It is 'possible'. However, you will have more trouble in finding a compatible and legitimate after-market PCI-e based SSD than you will actually replacing the current one. The only ones iFixit sells are used-genuine Apple OEM refurbished.
According to EveryMac:
As first reported by AnandTech, the "Mid-2013" MacBook Air models all use a PCIe-based SSD rather than a SATA-connected one like earlier SSD-equipped MacBook Air models. The "Early 2014" MacBook Air models are practically identical to their predecessors and use the exact same internal storage.
More specifically, AnandTech shared that the SSD in these lines use a custom Apple design with a "PCIe 2.0 x2 interface, capable of a theoretical maximum of 1 GB/s in each direction. At the moment, compatible SSDs are few and far between, so installation instructions effectively are theoretical."
The last sentence is the crux of the matter.
Essentially, not currently, but quite feasible in the future. The guides are all provided, it's just a matter of the hardware hitting the market.
There are no compatible third party upgrade solutions in the market as of the date of today, 16 August 2015. The only way is to buy from Apple or from companies like https://ifixit.com who sell refurbished Apple SSD storage.
The SSD drive in a MacBook Air is a custom PCI Express drive. A new 256GB drive will cost about $450,-.
You can order a drive and find installation instructions on iFixit
As Buscar says, warranty is gone when you open your machine. Ask the Applestore for a price and consider yourself what it wise.