Unless I'm misunderstanding what you're asking, the answer is surely no...consider for instance $\mathbb{C}P^n$ with its standard symplectic and complex structures.  This admits embedded J-holomorphic curves of arbitrary large area (take a high-degree curve in a plane $\mathbb{C}P^2\subset \mathbb{C}P^n$), and restricting to a disc within any of these curves would give you a $J$-holomorphic disc u of arbitrary large area (which I assume is what you mean by the symplectic volume of u).  

On more general symplectic manifolds $(M,\omega)$, the h-principle gives you immersed symplectic spheres in every homology class $A$ with $\int_{A}\omega>0$; these spheres can be taken embedded if $\dim M\geq 6$ and embedded away from finitely many transverse double points if $\dim M=4$.  In either case you could construct an almost complex structure $J$ on $M$ with respect to which an arbitrarily large-area subdisk of the surface is embedded and $J$-holomorphic.  (This is admittedly a little weaker than the first example, since here we're choosing $J$ after we choose the surface--so all it shows is that for any $C$ there is a pair $(u,J)$ where $u$ is a $J$-holomorphic disc of area larger than $C$, with $J$ possibly depending on $C$.)