If you have a 2017 or earlier model MBAir, then you can try disconnecting the battery from the Logic Board to power on the laptop using just the charger. If the laptop powers on, then you can try reconnecting the battery while the laptop is still powered on to see whether you can get the battery to charge. Do not power off the laptop or disconnect the battery until the battery charges to at least 10% or the battery may be permanently damaged. Just be careful not to let the exposed internal components to come into contact with any metallic or conductive object or surface or the laptop will be damaged.
Do not let the battery become completely discharged for any length of time or you will encounter problems powering on the laptop or even charging the battery. A Lithium-ion battery can become damaged when completely discharged for an extended period of time.
If you have a 2018+ model, then I do not recommend opening the laptop because the connectors are really fragile and easily damaged, plus it is more involved to disconnect the battery and difficult to remove the Bottom Cover. A 2018+ Mac will not run properly without a properly working battery anyway. I don't recall how well the 2017 and earlier MAirs behave when the battery is disconnected.
You should have Apple or an Apple Authorized Service Provider replace the battery for you while it is still an option. While Apple now offers to replace batteries for up to 10 years, the details of availability are not clear. I was just recently informed that Apple is still offering battery replacement for 2013 MBAirs (at least through an AASP who actually performs repairs in their shop) even though the laptop is considered "Vintage" for the last couple of years. The quality of third party Lithium-ion batteries is extremely poor even when purchased from a reputable vendor such as OWC, so I personally will chose an original OEM battery repair whenever possible.
