It takes more than wings to fly, just look at the poor penguins. The first problem is that simply weaving webs between a spider's legs would probably not generate enough lift to keep her airborne. Flying creatures have a specific body plan that allows them to fly, if you just add a couple of wings to a hippopotamus it won't be able to fly.
On top of that, the spider would not have the necessary muscles to enable it to flap these wings even if they were capable of producing sufficient lift.
While it is theoretically possible (I stress theoretically) for a spider species to adapt over time (long long time) into a flying form, then it would no longer be a spider. So, in answer to your question, no, it is not possible.