Root finding of an arbitrary polynomial with degree $\gt 4$ over any field is supposed to be a trial and error process, as far as I know.
If you are not dealing with any special type of polynomial, I can't see how any library is going to help you. Fast algorithms of finding roots for polynomial modulo p are there, for example here. You can also try Pari library written in C, which has implementations of this algorithm.
If you can factorize the polynomial into smaller polynomials of lower degrees so that you can search for roots trivially, you can use Chinese Remainder Theorem to combine the results and get roots for original polynomials.
If you have a special property of the polynomials you are using, you can exploit it. I will advise you to be a bit more specific about the problem you are dealing with, rather than posting same problem in different sites again and again.
I should have commented, but I don't have the necessary reputation yet. I'll try to update the answer as soon as I have a clear understanding of what you are dealing with. And please feel free to share your efforts so far in your post and comments. Thank you.
EDIT: You can also check out here for algorithms for finding roots of polynomials. There is always an option of using data structures provided by existing packages and mixing them as you see fit.