The Concorde, like many other supersonic aircrafts, had very thin airfoil mainly due to aerodynamic reasons. This answer gives a good overview on this topic.
Anyway the airfoil was definitely not flat i.e. neither a straight line nor without thickness. The following picture shows the central section of a Concorde and it can be seen that the airfoil there was quite thick:
Concorde central section, aft view (source)
Following one rib from the leading edge to the trailing edge, like the 33+51 in the next cutaway (source), one can more or less unterstand the airfoil shape:
Some reasons for the wing having a delta shape and not being flat can be found in this answer.