This one may prove a bit difficult.... [famous last words] York City - Kettering Town - Barrow - Havant & Waterlooville - Nuneaton Borough - ????? Most recent first
Crawley Town? - all made it through in recent years to the fourth round DRAW of the FA Cup, although not all played in the fourth round?
This is a difficult one, so I'll let on that to find the [missing] 6th team on the list you will have to go back to the 2002-03 season.
The missing side is currently in the Blue Square South, having finished in the playoffs but failing to gain promotion to the BSP
Ok this is probably the most difficult one I've put up so here's the next clue.... The most Recent (York City) is from the 3rd Round in 2009-10