Yes, the matter was put to vote. Well, at least sort of, and only in East Germany.

After the dramatic events of 1989 (weekly demonstrations of civil rights activists, the local elections (Kommunalwahlen) of May '89, which were for the first time proven to be fraud, and the fall of the wall in November), the German Democratic Republic's parliament (called Volkskammer, "people's chamber") was elected freely for the first time in March 1990 with a voter turnout of 93.4%. It was a landslide victory for the Allianz für Deutschland ("Alliance for Germany"), a coalition of conservative and centrist parties dedicated to market economics and quick unification with the Federal Republic. The Allianz campaigned with the slogan "Nie wieder Sozialismus" ("never again socialism"). The coalition was led by the East German CDU which was modeled after the West German conservative party CDU (which held the government under Chancellor Helmut Kohl in the FRG at the time). The CDU alone gained 41% of all votes which enabled her and her partners to form a majority in the Volkskammer and a stable government. This government (headed by Lothar de Maizière) proved to be the last administration of the GDR while simultaneously being the first to be elected fair and freely. The de Maizière administration began the gradual phaseout of the GDR and its admission to the Federal Republic over the course of the following year.

The fact that the East German populace in 1990 voted overwhelmingly in favor of parties openly opposed to a continuation of the GDR, can be interpreted as a vote for unification, I think, and was viewed as such by many. The West Germans on the other hand were never asked to cast their vote on the matter, but given the shared euphoria of events like the fall of the Wall, it is most likely that it would have turned out positively as well. (There were probably surveys made on the matter back then, but as I am in the train on my phone right now, I must encourage you to search for yourself.)

I wrote some more on this topic here where the OP was asking if there were significant groups opposing unification. If you are interested in more details, go ahead and ask! I will be able to answer tomorrow probably.

Hopefully LBo87's train trip went smoothly and can fill in the spots I missed (I'm sure there are some)

One of the major problems that faced the GDR was that it had to confront multiple crises in 1989 and its leadership was more reactive to events than proactive. The flight of tens of thousands of East Germans into West Germany created a crisis of legitimacy for the GDR. The out of control situation gave political reformers within the SED (the main political party of the GDR) ammunition to unseat its General Secretary Erich Honecker and replace him with Egon Krenz. Ironically, Krenz was Honecker’s own “crown prince” and was the Honecker’s own hand-picked successor. The problem was that Krenz had to deal with a series of broad-seated problems that were pervasive within the GDR and had a very limited time in which to accomplish these goals. The SED leadership in the Politburo had planned for a more organized and straight-forward series of gradual reforms, but events forced them to rush through and extemporize what was supposed to be a measured process that would retain the GDR as an independent political entity.

Adding further pressure to Krenz and the reformist SED was the increasing presence of the East German street calling for greater transparency, open elections, and a massive overhaul of the SED state. These demonstrations snowballed once it became clear that the 1989 was not going to repeat 1953, when the SED had sent in tanks to crush demonstrators. The general flat-footedness and incoherency of the post-Honecker SED is demonstrated by the series of decisions that led to the collapse of the Berlin Wall. On 9 Novemeber 1989, there was a live press conference held by the SED’s propaganda chief and Politburo member Günter Schabowski and an Italian journalist asked him a question on the new travel restrictions in the GDR. Schabowski, who was out of the loop for the Politburo meetings on travel restrictions, improvised an answer that asserted that the GDR was going to implement regulations allowing GDR citizens to leave any of its border crossings and when pressed, he indicated they would be immediately implemented. This led to a massive influx of Berliners congregating at the Wall and the border guards simply opening the gates when they could not receive any clear orders to do so.

The impotence of Krenz and the SED gave Helmut Kohl a golden opportunity to pursue unification and he and the CDU took it. This was actually not as easy of a process as one might assume. Both France and the UK were highly skeptical of German unification being within the larger interests of Europe. Part of Kohl’s canniness was that he recognized that the SED’s impotence gave him much greater latitude on the international stage. Kohl had developed a degree of contacts within both the USSR and the US and was able to sell both Bush and Gorbachev on the idea and popularity of German unification. Kohl gambled that if put to a vote, the citizens of the GDR would vote for unification. As LBo87’s answer notes, this gamble was correct and Kohl was able to present German unification as a fait accompli.

It should be noted that the SED never intended for the GDR to dissolve and hoped that it had enough time to reform itself. Their perception was that the SED would be able to navigate into a "third-way" that would create a reformed socialist Germany. This was not an entirely illogical assumption. The idea of German unification had been something of a dead issue since Brandt’s Ostpolitik had formally established a degree of recognition to the GDR. Unification had to pass over the significant stumbling block of memory of the Third Reich and the spectre that German unification would lead to a renewed German drive for hegemony. This was a popular concern within most European states and built on both right-wing nationalist fears and some among Europe’s left-wing that perceived the CDU as possessing strong ideological continuities with the Third Reich (the logic here is that it Adenauer’s soft denazification policies had led to individual Nazis taking up positions of power within the FRG’s government and industry). Kohl was able to push for unification and twist or spin votes against the SED as a vote for a swift unification. While there was a sizable majority of East Germans who were for unification, there were quite a few East Germans who were also quite upset over their stagnant economy and hoped that a vote for Allianz bloc would economically integrate East Germany into the West and recreate the Economic Miracle of the Adenauer era.

Sources

Jarausch, Konrad Hugo. The Rush to German Unity. New York: Oxford University Press, 1994.

Kotkin, Stephen, and Jan Tomasz Gross. Uncivil Society 1989 and the Implosion of the Communist Establishment. New York: Modern Library, 2009.