Feed stores or Jefferslivestock.com for the Corid or Sulmet... I hear the Corid is a better choice, not as hard on the chickens systems. (Sulmet is Sulfa)...
Best of luck... I am sorry she is so sick. Use the search here on BYC for "oral dosing", "giving oral medications", "giving water and food orally when they won't eat or drink", etc.
There are pictures and instructions about how to best avoid the windpipe, the airway.... the hole that is right behind the tongue.... you want to be careful not to get any food or liquid into that hole. Aspiration of water, medicine or food will cause quick death, from what I have heard.
I have given worming medicine (Valbazen) with a syringe (no needle) into the open beak, to the side and back ... behind, down, to the side of the tongue and airway... no problems yet. It was "difficult" because I was scared the first couple of times, now it goes more smoothly.
In older chickens, not so young... these symptoms could be an indication of worms.... if a chicken is over, say, 6 months old and had these symptoms... it could be either.. but, worming would be recommended as well (not simultaneously with the Corid though), and probably first. Just not for a chick so young.. too young to have worms.
Sulmet is not recommended for laying hens at all, ever... so... I am not sure of the withdrawal period if it is given, I think most wait three weeks after treatment. It does go into the drinking water and there are directions for each disease it treats on the bottle. I do not know how much you would give, when it is diluted in water, if you were having to give it orally to a chicken who won't drink.
As far as I know... the local feed store is the place to get all your antibiotics, worming meds, feed, etc... but, when you ask for things "for chickens" and don't have the medication name... often you don't get what you need... as we use many things "off label" to treat chickens.
Others know more than I do... and the Search here on BYC is very helpful.
Best of luck... I am sorry you don't have a well stocked feed store nearby... take care.