Starting in 1801, gray great-coats (greatcoats) were issued to all non-commissioned officers and privates[1]. A very excellent article detailing the great coat as issued to British troops from 1801-1815 is here (http://www.warof1812.ca/greatct.htm). I highly recommend reading the article.
The following is the list of approved vendors. Please do not purchase a greatcoat from elsewhere without officer approval. Once you have received your greatcoat, it will need to be waterproofed. The article linked above lists the proper period correct procedure.