UV-Vis is probably the last thing I would use to find out the refractive index of a material. There is a way but the error could be inacceptable. Simplest approach- you produce a smooth layer of your polymer on a black, strongly absorbing substrate and measure the specular reflectance. The reflection of light should be mainly due to a Fresnel reflection on air/polymer interface and could be calculated back to produce an estimate of refractive index. Altenatively, you could try measuring a layer of your polymer (provided that it is absolutely transparent and does not scatter light) on a transparent substrate in transmission, however, in this case you deal not with one but with three interfaces air-polymer, polymer-substrate, substrate-air, which makes the problem much more mathematically involved. There are methods like T-matrix, that you could use to solve it but if you have a choice, you should better go for something like ellipsometry. UV-Vis is not really suitable for your problem.
Ashraf, your question has a complex answer and I will add to the detailed reply from Roman. I have measured refractive index of polymer films using an Abbe Refactometer (details and images can be found from a Google search or Wikipedia); I used a bench type though their are simple hand-held types. The instrument is designed for liquid films, however polymer films that give good contact with the prism are suitable. Refractive index is dependant upon density. The result will vary with crystallinity of the polymer: semi-crystalline polymers are birefringent due to the two phases, however two refractive indexes will not be revealed with the technique discussed here. Plasticizers or other bulk additives will contribute to the measurement, including fillers though the method will not give an observation as opacity of the film increases. I used a refractometer to measure ultraviolet initiated polymerisation kinetics of acrylate esters, the refractive index of the monomer transforms to the refractive index of the polymer.
The answer of your question is definitely "yes". There are few ways, here I еxplain some of them:
1. If you prepare the polymer in thin film form with thickness of approximately 1 micron than you can use Swanepoel's method for determination of n, k and d simultaneously from interference maxima and minima in transmittance spectra. However, have in mind that you have to use glass substrate with higher refractive index (n=1.7, 1.8) than ordinary optical glass (n=1.52). Otherwise the optical contrast of polymer and substrate is very low and the peaks amplitude will be very weak. Even it may happen the peaks to be indistinguishable.