It is fairly easy to find a replacement stair lift battery. Most battery shops can supply replacement batteries, I prefer Batteries Plus in my area.
The bigger question is should you replace them yourself? Some manufacturers require battery replacements by qualified engineers, or they may void the warranty. It is easy to replace the batteries on some models while other are a bit more complex as the require the removal of the carriage cover and safety edges.
Unfortunately there is no guide that will describe how to change the battery for every manufacturer or model, and there is considerable risk in doing it yourself. Risk for both the usability of your stair lift, the welfare of the stair lift user, and the welfare of the person replacing the battery.
It can be done by a person with the right technical know how however you are probably better off paying a qualified technician to do it.