Great question. The short answer is: "not much of one." The longer answer is: "it depends on where/when one is talking about, and how serious one cares about the definition of 'role'."

The modern research university is a very modern creation, something that doesn't really exist until the 19th century. Medieval and early modern universities were about educating the clergy and to a lesser extent future courtiers. By the 16th and 17th century they were ever more educating courtiers — sons of wealthy men destined to be sort of future bureaucrats. They were not sites that were dedicated to pushing the frontiers of knowledge, at least not in the way we would later think about it. They had a reputation, in their day, for being places where people went to bicker about philosophy, essentially, and were considered by many of the forerunners of the Scientific Revolution to be authoritarian in nature, about rote learning rather than research. There were some major figures who were housed in universities, notably in England during the later part of this history (e.g. Newton), but the university in this case was less an engine for his work than a place where he was allowed to work on what he wanted without too much interference.

To put into perspective how un-useful universities were in this period, note that Galileo's big triumph and motivation for dedicating his early discoveries to the Medici family was the hope that they'd let him leave his university position and instead become a "private" researcher for them. At the university, Galileo's research interests were not at all supported and he had to spend a huge amount of time teaching. Working for the Medicis, he could dedicate himself to research and had access to much greater resources.

If one looks at the Scientific Revolution (a nebulous concept, we should note), very few of the early innovators were associated with universities. Most relied on some kind of other patronage, such as the Church (Copernicus) or wealthy families or sovereigns who liked supporting intelligent people (Galileo, Kepler, many others). Some of them were themselves just quite wealthy to begin with, as well, and were able to afford their own research, or had it essentially subsidized by the "state," which is to say, other nobility (e.g. Tycho Brahe).

The main place you start to see universities playing a role early on is England in the late 17th century. How much of that is related to the specifics of the English context at this point, I don't know, but it's basically the only place where you start to see a lot of people associated with universities also doing important work, and institutions like the Royal Society of London being closely associated with academia (e.g. Gresham's College). In most of the other sites of knowledge growth in Europe over this time, my feeling is that this was not the case, though again at this point each context is so unusual and different from other another that drawing any big conclusion is difficult (e.g. France, "Germany," the Netherlands, Russia, "Italy"... these are all really different at this time, and the ones I put in quotation marks were not unified states at this point, adding to the complexity).

Again, the modern research university, where the professors are meant to be doing some teaching but also some research, doesn't really arise until much later. The Humboldt Plan in the early 19th century put research on a strong footing, and was eventually exported to a lot of places outside of Germany. What we think of as the modern university basically comes out of this approach, and in many places was not explicitly adopted until even the early 20th century.

The Scientific Revolution is a tough thing to generalize about because it's not clear when/where it was, if it really happened at all, in the sense of "one big shift" and not "a lot of little shifts that happen over centuries." But I think it is fair to say that universities did not play a big role in making it happen. I am not sure one could say they hindered it, except as much as a place for smart people to talk and work, they didn't really encourage it for the most part, and in that sense I guess you could think of it as a form of hindrance. Some of those involved with the Scientific Revolution saw them this way, feeling like they got in the way of real research. This seems a bit like a kind of backwards reasoning to me; I would prefer to say they just didn't play much of a role, compared to other factors (you could make a bigger argument for postal services in encouraging the scientific revolution than you could for universities — without reliable communications you really don't end up with a "community of letters," or the idea of journals and peer review, and things like that). They did provide a context for some of the people who participated in this work, and that counts for something. But they were just one of many contexts.

A great overview on the Scientific Revolution is Shapin's The Scientific Revolution, and I am fond of Peter Dear's Discipline and Experience as another way of thinking about what kind of "Revolution" took place.