Is there an alternative to Time Capsule to do a wireless time machine backup. My mom already has a router and doesn't want to replace it but I still want to make sure she has a backup. She never has her MacBook connected so I'm looking for a wireless solution.
You can still use a Time Capsule, just don't use it as a router. You can turn it's routing off by putting it in bridge mode (using the AirPort utility) and simply connecting a network cable between it and the existing router. You can turn wifi off as well if you don't want to use it, networking will happen via the existing wireless network.
As a somewhat cheaper yet officially supported alternative, an AirPort Express with an external USB drive works too. If the backups aren't too big, an older model works too, but since it's slower, big backups will take a while.
Alternatively, any NAS with "Time Machine Support" works out of the box as well, QNAP and Sinology make those for example.
Why don't you buy any (cheaper) NAS drive? The only requirement is to have a Time Machine Service, which most of them have. (Time Capsule is also a NAS, but restricted to Apple devices.) You can plug it into her router to do the basic setup. You should see it then as a valid Time Machine target drive.