Is it possible for me to buy the rental card for ¥300 at the Hangzhou railway station?

There is no place to buy such kind of service near the local railway station.
You can take the local subway line 1 to from the railway station to reach Longxiangqiao Station. Then you can follow the direction board to the IC Card Center to find the place to get a Z Card. From the railway station to Longxiangqiao, it takes around 10 minutes.