Short answer: yes, but the highest MW measurable is pretty low. The controlling equation for dilute solutions is Delta Tf = Kf b i ; where Kf is a constant for the solvent, b is molality and i is number of particles (usually 1 for a polymer).
For example, if you could dissolve your polymer in camphor (MP 179.8, Kf = 39.7 - one of the largest Kf values) and you were able to get 10 mg into 100 mL. For MW 10000 that is 10^-5 molal so the Tf only changes by 0.0004 C - very hard to measure. If you could push it to 100 mg (1 mg/mL) and the MW was only 1000, then the Tf changes by 0.04 C - just approaching measurability.
The only colligative property that gets you into the 10000 - 20000 MW is vapor pressure osmometry.