Hi Caroline,
You can certainly use your injury to point to problems caused by your spinal issue. However, the injury itself is unlikely to be considered when you are assessed (you need to have had a problem for at least 3 months and it must be expected to last for a further 9 months in order for it to be considered for PIP).
If you feel that a wheelchair would help you get to the assessment, you could consider hiring one from the British Red Cross if your doesn't look as if it will arrive in time.
May I suggest that you bookmark/favourite this on your web browser now so that you can find it easily in future? This will allow you to return with further questions or comments about your PIP without having to start a new topic each time. We ask members to keep everything relating to the same claim in one topic as it helps us enormously - and I hope you will find it useful too