For any $F$ which satisfies conditions to be described below there are precisely two quadrilaterals  $ABCD$  which have area $F$ and side lengths $a,b,c,d$.  To be precise, I am using directed areas (i.e. $A\wedge B+B\wedge C+C\wedge D+D\wedge A$) and side lengths are $|AB|=a$ and so on cyclically. Hence the required max and min are the largest and smallest values for $F$ which satisfy the given condition.  

The condition on $F$ is the positivity of  an explicit  sextic polynomial whose coefficients are functions of the side lengths.  Hence the required optimal values are roots of this polynomials.  The polynomial is, in fact, a cubic in $F^2$ so that it can be solved by radicals.  The concrete form of the cubic is too complicated to quote here.

Preliminary result or rather conjecture since it is provisional until double checked:  
the two areas are $$\frac 1 {16}(-a^4+2 a^2b^2-b^4+2a^2c^2 +2b^2c^2-c^4\pm 8abcd+2a^2d^2+2b^2 d^2-d^4).$$

The plus sign gives, of course, the maximum.