Is it possible to estimate the location of aerodynamic center without experimental moment data? In airfoils, the aerodynamic center is usually found close to the quarter chord point, but for quick calculations, we assume it lies on the quarter chord. 

What I am looking for, is if there is any way to find a rough estimate as to where the aerodynamic center is actually located on a tested airfoil - given only lift and drag data.

The tested airfoil is mounted on the sting on its quarter-chord point, which sits on top of a force transducer. The angle of attack is varied throughout the test.
The pressure distribution $c_p$ on an airfoil is necessary to determine the aerodynamic center of an airfoil. From the pressure distribution you can obtain all three coefficients $c_l,c_d,c_m$ (well, for accurate $c_d$ you better have a rake for high $L/D$ airfoils). Having only the first two, as in your case, is not enough. Simply because there are many $c_p$ distributions that will give the same $c_l$, $c_d$ forces and different $c_m$ forces. These airfoils will also have a different aerodynamic center. Thus, the answer is no. If you don't have means of measuring the moment, couldn't you just vary the mounting position of the airfoil to estimate moments and do your job?

However, in case you want only a rough (experimental) $c_m$ estimate you can relate your airfoil basic characteristics ($c_l,c_d$, zero lift angle, camber and thickness distributions etc.) to similar airfoils found in this classic reference and interpolate.