S.:
I feel your pain, I have been there myself. I was placed with my then-infant niece over two years ago, I adopted her last summer. I have also been a foster parent and have dealt with LCCS in the capacity of a CASA. Needless to say, I understand how difficult they are to deal with.
I see that your heading is "temporary custody," and I'm assuming that you have temporary custody of the children. Unfortuneately, once a family member takes temp custody, LCCS has little interest in providing any further assistance to them. Since the children are not in LCCS custody and not in a foster home the agency doesn't feel that assisting the children with necessities are their responsibility. Although agreeing to take temp custody of the children probably seemed like the most natural thing in the world to you (and maybe it even made you feel more secure in having them), it was probably the worst thing you could have done if you needed assistance - and LCCS knew that but still probably told you that taking custody was a good thing.
My only advice at this point is to continue to make a racket and to make it as far up the chain of command as necessary. If you can, leave a paper trail - send letters to the case worker and cc them to the case worker's supervisor, the unit supervisor and the director, Dean Sparks. If the children have a CASA or a GAL (guardian ad litem) make sure they know of your problems as well as they are there to look out for the children's best interests --and getting diapers surely falls into that category.
You may also want to contact Kinship Navigator if you haven't already. It's a program run through the Area Office of Aging that's devoted to assisting families who have kinship placements such as yours. I've never really utilized their services but they offer monthly support groups, respite care, and referrals to services. At the very least I'm sure you will find them to be a group of people sympathetic to your plight and possibly able to help with services.
If you are really in need of diapers and have no other way of getting them, a church I have attended occasionally has a toiletry pantry one or two Saturdays per month - they are always looking for diapers to stock it so you may be able to get some there. The church is First Unitarian Universalist, it's located at Bancroft and Collingwood in the Old West End. You may also want to check garage sales and Ebay for "leftover" diapers people are trying to sell because their children out grew the size (not my first choice for getting diapers either, but diapers are diapers no matter where they come from).
Hope this helped. Please contact me if there is anything else I can help with.