Within my suggestion you can do this
=SUM(SUMIFS($D$1:$D$27,$B$1:$B$27,"May",$C$1:$C$27,{"FEC","ORANGE"},$A$1:$A$27,{"NY";"LA";"GA"}))
Note that with two "array constants" one has to use comma separators, one has to use semi-colons. If you wanted a third then that would be problematic using that syntax - another approach which is more easily extendable to large numbers of "OR"s is to use SUMPRODUCT with ISNUMBER(MATCH, i.e.
=SUMPRODUCT($D$1:$D$27,($B$1:$B$27="May")*ISNUMBER(MATCH($C$1:$C$27,{"FEC","ORANGE"},0)*MATCH($A$1:$A$27,{"NY","LA","GA"},0)))
In that setup the separators can be commas or semi-colons, it doesn't matter which (as long as you are consistent within each part).