I think it took roughly eighty years, from the end of Prohibition in 1933 to the effectively full-on wealth-control of the US Government, with legalized bribery (through SuperPacs, the Supreme Court decision allowing them occurred in 2010).
Stephen Colbert detailed the real-life rules of the SuperPac disaster on his comedy show (with real lawyers explaining it) and showed in detail how SuperPacs now allow politicians to personally pocket unlimited campaign contributions, tax-free no less, and do NOT have to disclose donors, amounts, etc. This allows legalized bribery in unlimited amounts to any politician with a few hundred bucks to set up a SuperPac.
Since the end of prohibition, there has been an increasing tendency of politicians voting to help provide legal cover to whichever criminals gave them enough money, and effectively extend the "privilege" of the very rich (even if criminally rich) and politically generous to be nearly immune to prosecution for any crime short of murder (and by now including crimes involving what once would have been considered murder). There has been a gradual increase of that for decades. Glenn Greenwald documents much of this in his book, With Liberty and Justice for Some.
I think your story can follow a similar dynamic; smart criminals and corporatists will not "take over" the country by force, they take over the country by paying a back-door tax, far less than what their income tax would be, but they use a few % of their profits to help elect politicians, from the ground up, friendly to their financial overtures, and will spend big to slander, compromise, frame and defeat any candidates that are NOT friendly to their overtures. Once they own a majority, they have control.
"From the ground up" means they finance the careers of police chiefs, city councilmen, mayors of small towns, state level legislators, governors, as well as national congressmen, senators and presidents. Half of this is to get these people to give them multi-million (even multi-billion) \$ contracts and business (as you suggested), knowing they (meaning the politicians) will get their fair cut of the millions, or "favors", that can range from sexcapades with high-end call girls to sweetheart "investment" deals in real-estate (a "distressed" seller sells the politician a \$250K house for \$150K), "used" cars (brand new cars bought and sold to the politician for the minimum legal price), hot stock market tips (when the fix is in), even hard to trace "gifts" (alcohol, entertainment systems, house re-modelings, vacations, etc).
Sometimes these bribes the form of after-the-fact jobs (lobbying positions, business partnerships requiring no investment from the former politician, "consultant" positions, etc). The bribers are consistent in these payoffs, even after the politician can no longer directly help them, because it is important for other, sitting politicians to see the briber takes care of their "helpers" once they leave office. These jobs and sweet deals are a form of advertising to still-active politicians.
Note they don't have to do this nationwide, they can start as small as a town, to start the public-money tap, and aggressively re-invest the profits into expanding their influence into the cities, then the State, then neighboring states, like a cancer, all the while collecting politicians that steadily increase the amount of public money flowing into their corporate bank accounts.
An important part of this scheme is to explicitly choose to support corruptible politicians. This can often be discerned by doing comprehensive background checks into their character.
This is important because it lets the bribers develop sticks to back up their carrots: Audio and film evidence of the politician doing compromising things, in particular sexual things like enjoying the services of prostitutes, but also things like getting illegally high, or getting drunk, taking explicit bribes, trips, gifts, or actually committing felonies. Just in case the bastard ever grows a conscience or balks at something particularly heinous; the bribers will want to show them a private screening of a convincing story.
If your Security company begins in today's legal landscape, and they are smart and ruthless, they could plausibly take over in 30 years or so. But beware, they will undoubtedly encounter resistance from the current owners, and I would not be surprised if it were lethal.
In America there were a lot of laws to be circumvented to engineer our current legally-sanctioned corruption (IMO obviously, and talking about both parties), so it took about 80 years.