In his role of preaching hope and encouragement to the Babylonian exiles, the prophet Jeremiah (Chapter 29) gives specific instructions on how Jews are to live among their “hosts.” He tells them to settle in: build homes, plant gardens, marry and grow families. He also adds that the exiles should “seek the peace of the city where I [meaning God] have caused you to be carried away captive and pray unto the Lord for it; for in the peace thereof you shall have peace” (v. 7). “Seeking the peace of the city” has historically been understood as participating in the welfare of the countries in which we have lived. Voting in elections can certainly be construed as participation in the democratic process that best assures the welfare of society. While there is no legal obligation to vote, there is indeed a civic obligation and, if not an actual Jewish legal obligation, voting is certainly interpretable as a prophetic instruction.
If you have read the answers of my two colleagues, you will find that they have each laid out excellent responses. The essential feature of both is that they find no specific Jewish legal obligation (under Halachah) requiring one to vote. They each give some good support to the reason one might feel such an obligation as a civic duty, but they do not offer a Halachic reason. It seems to me that they are correct, because I can neither think of nor find any Halachic obligation that would direct one to vote in civic elections.