Is it possible to estimate the location of aerodynamic center without experimental moment data?
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.