Hello. I'm sorry to hear about this situation. Please know I'm not trying to be rude or hurtful here and I may be way off base here but here goes.
What's the issue here? Is this about money? Or is this something deeper? You are fortunate enough to be a SAHM - it's $40. I know it can go a LONG way. Is it the principal of the matter or what? You knew when you married your husband he had another child. Divorce is rarely ever pleasant and when children are in involved, it can sometimes get VERY nasty. It's NOT the child's fault.
In my opinion you have every right to be angry. However, you have a choice to make - to be the responsible adult or the whining, sniveling "new wife". I would chose to be the adult.
This child's mother is obviously irresponsible - if she's not going to school, she should be working full time - but that's my opinion. She shouldn't have a live-in boyfriend either - she has an example to set for this girl. Remember this child is observing EVERYTHING in her world - she'll be mimicking it soon enough - which world do you want her mimicking - yours or her biological mothers?
It's obvious you do not like this woman. However, it IS obvious you love this child! I hope you don't show any ill-will when you talk on the phone with her, etc. as the child will be listening, watching and picking up.
I was a step-mom before. However, I fell in love with my daughter (I met her dad when she was 9 months old and her parents were separated) and we gained sole-custody of her. Her biological mother never worked and it seemed that "E" never had clothes that fit her, diapers, etc. even though my then-husband was paying her child support.
My daughter's biological mother lived in a different state, we paid for her to come to a birthday party. She ALMOST didn't make it.
Your step-daughter, obviously needs guidance, love and support from a woman. That should be YOU. Do NOT EVER say anything negative about her mom in front of her or if there is a chance she can hear it. Be polite when you pick her up and drop her off, etc. NOTE EVERYTHING (lack of responsibility, part-time employment (especially if she's NOT going to school, condition of home environment, lack of financial responsibility, live-in leach of a boyfriend, etc.), lack of taking child to Dr's, etc. and get custody so that YOU can give this child a warm, loving environment.
You are fortunate enough to be a SAHM. You have an opportunity to make a HUGE difference in this girl's life. Suck it up, pay for the supplies and everything else she needs (note I said NEED not WANT - you will be teaching her responsibility and many other things as well) give her your love generously.
DO NOT complain about buying the supplies - she will end up respecting you and following your lead and example - which is what I would hope you want - for her to be a successful member of society rather than a leach.
Being in your situation is NOT easy. It's all about attitude. A child's welfare is at stake here - it's not about the school supplies, it's about a child who needs your support. I would fight for sole custody so hard - a child deserves and needs a good example, love and support.
I'm sorry this is long and drawn out. I truly don't believe this issue is about who pays for what. Any animosity you harbor towards this woman isn't going to help you or the child. Set a great example for the child. Give her what she needs.