B.B.
I am a licensed provider and although you probibly know this already but for those who don't... The reason a lot of people won't accept what is refered to as "part time drop ins" is for planning sake. If I have 6 spaces open (max a single person can be licensed for with no helpers) I want to fill those spaces with full time kids on a schedule. I can charge full time prices and schedule more fun stuff since I know what ages, abilities, and likes I have thru out the day... Its business. Drop ins are the unplanned here and there kids.
I may be an exception to the rule... I do not fill all my spaces on purpose. There are those of us out there. I have 3 part time kids (after school)... Full time during summer...
Get ahold of your local 4C office. They can give you a list of all the providers in your target area that have openings and accept part time/ drop ins.
Legally, (I hear they are in the long drawn out bureacratic process of changing the current laws) NOONE can care for an unrelated child more than (If I remember correctly) 5 days a YEAR with out being licensed... But that does not mean much. :-) Ask your neighbors, friends, etc. There may be a grandma out there somewhere that would love a few extra bucks and cuddles from a munchkin... BUT if you are accepting or applying for childcare assistance then the person needs to be licensed.
Keep calling around and you may stumble on an exception. :-)