Marian Reforms?
The "Marian Reforms" aren't neccessarily considered to be that distinctive a change, nowadays. It used to be that historians gave Gaius Marius the credit for turning the Roman army from a militia of small-time landowners into a professional force of urban proletarians, thereby solving Rome's endemic manpower problems, but this view is something of an oversimplification.
For one thing, modern evidence suggests the Roman army didn't actually have that big manpower problem in the late 2nd century B.C. The losses of the Punic Wars were severe, but Italy recovered quickly, and afterwards the population remained steady. Nor is it really believed that the peasant class was eroded by soldiers being forced to serve for long campaigns abroad: most recruits were younger, unmarried rural men, for whom there wasn't enough work anyway and who would welcome the chance to win glory and booty in war. (Well, they welcomed it if they got to fight in the rich Greek east. Nobody wanted to go to Spain to fight the hill-tribes there.) Unless there was a great crisis, there doesn't seem to have been much resistance against or difficulty with conscription.
What happened was a more gradual evolution of the Roman army, in response to their empire changing from a city-state to an entity spanning the length and width of the Mediterranean. Marius played a role, but perhaps more to confirm and codify practices that had already been getting more common. The changes that took place around the turn of the 1st century B.C. were significant, without a doubt. Property requirements for enlistment had been steadily eroded for decades, but Marius abolished the requirements altogether, and also took steps to standarising organisation and equipment. The role of the Italian Allies disappeared after the Social War, (91-88 B.C.) and soldiers came to see their service much more as a career, getting their pay, clothing, equipment and retirement fund from the state... or their general. These more permanent legions had a stronger identity, but it was one that was often tied to their general. We see several armies in this period that even took their name from the general leading them. (The Fimbriani, the Sullani) For many of the men serving in these units, the army wasn't a sideline, but it was their way of life. Although we shouldn't overstate these changes: soldiers may have served an average of 8-10 years, but it's a long way from the 20-25 of the Imperial army, and legions continued to be disbanded when the war they were recruited for was finished. Additionally, while most troops were volunteers at this point, conscription did continue.
The loyalty of the army
Now, the army certainly played a very important role in the cycle of civil wars and general chaos that finally ended up breaking the back of the Republican system of government. (Although as Publius Clodius or Lucius Appuleius Saturninus demonstrated, armies weren't an absolute requirement for chaos and civil strife.) The armies' willingness to follow their generals even against fellow citizens and soldiers was a crucial factor.
However, it has to be said that the Roman Senate played its own part in creating this situation. They absolutely refused to organise any kind of standard reward or pension for discharged soldiers, and their generals had to fight time and again to get laws passed that would grant their soldiers the land they needed. Without their generals, these soldiers from poor backgrounds had an extremely uncertain future. Add in the bonds forged by shared danger, hardship and glory, as well as the fact that most soldiers were recruited from remote and poor rural regions in Italy and did not feel that strong a connection to the city of Rome itself, and it is not all that surprising that their loyalty went to the man who led them, rather than the notables ignoring them back in Rome.
Still, that loyalty-to-the-generals thing shouldn't be overstated. There are also cases of Roman soldiers refusing to fight eachother, such as during the struggles between Marcus Antonius and Octavianus. And those Roman generals that did march on Rome or on other Romans always had to justify it to their troops, with appeals to injustices done to them or their honour, or to the soldiers themselves. Particularly notable is Caesar's use of two Tribunes of the Plebs, who had been injured in a riot in Rome while doing their duty. They returned to Caesar's army in northern Italy, where he paraded the wounded tribune (a legally inviobale official with great symbolic value, as they were supposedly the protector of the common plebeians, though by this time they were mostly acting in a different role) to instill outrage in his soldiers and convince them they have the right to march on Rome and restore order. You could even try to make the argument that Caesar's army (or at least some of the soldiers within it) were marching to save the republic, not destroy it.
The Late Roman Republic: Transformation or fall
So, did the changes in the army influence the transition from the Republic to the Principate? They likely did, but I'd suggest it is more the case that both changes were influenced by the same thing, already referred to above: the Roman Empire had gotten too darned big to govern as if it were a city-state.
Just as a part-time militia army wasn't really suited to campaigning in Armenia or Syria, yearly elections and a political system carefully designed to minimise the possibility for change and reform weren't really effective for governing an empire spanning thousands of miles. The republic ended up sticking to the old ways for the governance of Rome and Italy, and just delegating governance of the provinces to former magistrates, who for the most part were equally hunger for power, wealth and glory. This led to even further expansion, as the governors were always seeking wars so they could earn a triumph and acclamation back in Rome. These wars made a select few of the Roman elite extraordinarily rich and powerful, which created divisions and damaged the cohesion of institutions such as the Senate. It also led to hardship for the provincials, and sometimes to unrest, as either the Governor or the privatised tax-collectors known as publicani —or both— would extort them and drain them dry.
Obviously, much more could (and perhaps should) be said about the state of the late Roman republic than I can go into here, and I should note that there's a considerable division of opinion on how sound or unstable the republic was at this time. As ever, historiography on the matter is divisive. There are those like Brunt who say the Republic had lost the faith of all its driving figures and was ready to topple no matter what, those like Gruen who say it was essentially sound until civil war happened, and all kinds of positions in between. I'd disagree with the latter argument, but even so there's a difference between saying that the Roman Republic as it was could no longer effectively govern its empire, and saying that monarchy was the only viable alternative. Perhaps reform would have been possible.
So how significant was the post-Marian army's contribution to the eventual transition to the Principate? That kind of question is always extremely difficult, and the answer you arrive at largely depends on how you interpret the state of the late Roman Republic, as outlined above. It's certain that the army played a big role: just look at all the civil wars. But it's impossible to say whether the civil wars could have been avoided if the senate had taken a more pro-active role in regulating the new army, or if they had reformed their governmental institutions to better suit the administration of a vast empire, or if they had found ways to maintain the loyalty of the elite noblemen leading these armies. Perhaps a monarch would have arisen anyway. Perhaps a different system could have emerged. Perhaps the entire republic could have fallen apart into an "Age of Warlords" such as we have seen so often in other times and other places. This nearly happened after Caesar's assassination, after all.
In the end, I'd answer your question by saying that the post-Marian Roman army played a very significant role in the process of the transition from Republic to Principate, but many other factors were also in play, and it's difficult to say how significant a cause the changes in the army's structure were.
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