Take it from someone who has experience of agoraphobia - your mum must feel isolated enough without having to worry how it affects you too. It sounds harsh but I think your mum really needs time to focus on herself and maybe by walking alone, she has the chance to do that. She needs to heal and figure out what she wants and how to achieve it, and only she can do that.
We all have crosses to bear and I think you're being selfish by saying its her fault that you have no people skills because she doesnt take you out-ask your dad or another relative if you can go out with them when they go. If your mums as bad as you say, they must have noticed and will want to know how to help her.
If you go to school, you must be talking to people there, and there are teachers, school counsellors you can speak to.
Agoraphobia and other similar disorders are extremely scary and dibilitating for the people they affect, and all too often people, especially family members, dismiss the fact that theres anything wrong with them and just ignore the person whos suffering, solely due to ignorance and lack of information about whats wrong with their loved one - not always their fault although information is getting more widely available these days.
Your mum needs your support. Maybe by doing that you would develop a lot more people skills that going out to the shops with her would ever give you.
This world is about give and take - please learn to do both for the benefit of everyone