The simple answer is yes, you can calculate the molecular weight of a polymer.
On the other hand something like GPC (gel permeation chromatography) would give you an accurate figure of the polymer. If you have GPC, may be worth trying that for polymers. It can also give you info on distribution, poly dispersity etc. which are quite useful for determining the polymer molecular weight.
Many answers are already given. If your polymer is insoluble, MALTI-TOF can be used. Other methods are there for soluble polymer including GPC, GC-MS, Membrane osmometer, Vapor phase Osmometer, light scattering wtc.