Here are a few ways that I can think of :

• Buy Controlling Interest in Critical Supplies or Services
• Examples : food, helium, rare earths; if your company can become the largest provider of goods (and especially if your company provides those goods at a loss, so that there are few alternatives to using you) you have some leverage.
• Service Examples : negotiating right-of-passage with countries whose airspace the city flies over, being the "ferryman" to-and-from the surface, being the gatekeeper of information (mail, news, internet, radio, or other communication)
• Buy Controlling Interest in the Entity that Operates the City
• There may be a dominant industry in which the citizens participate. Could be a monastery, grand council, or a company.
• Your company could gain a foothold in this industry by :
  • Purchasing stock,
  • Issuing loans,
  • Grants
• Whatever the mechanism, your company will gain in exchange an increasing number of voting representatives to this decision-making body that your company may select.
• Having representatives in the government also allows your company advance knowledge of crises, scandals, weaknesses, and worries. Your company can exploit this insider knowledge to create new problems -- which the company would happily fix in exchange for more representation, or fewer constraints on their ability to operate.
• Buy Controlling Instance in the Entity that Owns the City
• If the ownership of the City is recognized and protected by any government, being able to own (or leverage) the owner of the city would allow your company to pressure changes to the operation of that city.
• Your company might be able to do this financially (as outlined in buying controlling interest in the City Operator), or by legal action in whatever legal body the City is recognized.
  • Examples of Legal Action :
    • Claiming the City as part of some inheritance
    • Claiming the rights of some co-founder of the City as co-owner
    • Claiming the City as payment from the Owner for some tort, criminal behavior, unpaid taxes, or similar
    • Some law that calls a floating city abandoned property
• Corrupting Officials
  • Your company might bribe some key people (shipping on-and-off the City, power, maintenance, police, computing), or Influencers (people who most of the citizens naturally follow, regardless of the on-paper power structure)
  • From this perspective, your company can exploit legitimate (but solvable) problems to create a popular uprising. See cultural warfare, below.
• Cultural warfare
  • The formula is tried-and-true : have your influencers remind citizen constantly about the Problem(s), blame every other problem on the problems, build up the citizens' sense of injustice to the point the citizens start committing injustices themselves, spread rumors of armed rebellion in the "right" social circles (the governments) - encouraging the government to overreact and create more injustice. Repeat until revolution, or detected. If detected, burn a scapegoat and continue with a new one.
    • Example : "There's not enough potable (drinkable) water to go around. This problem is one the City can't fix -- there's only so much weight the floating City can carry, and it only feeds so many citizens. Well, not enough drinking water is why Matt passed out at work yesterday. I heard it in the bar. And the Council, they don't do anything about it. You know, it looks like those Council folks have never missed a bath. Maybe, if they're not going to do anything about it, we should..."
    • When revolution happens, as the influencer of the Influencers, your company gets to move in and take charge.
  • Alternatively, you can saturate the City with your message. It's baked into every paper they pick up, every book they read, every movie they watch, every song they listen to. Gobbels used this with great effect to turn a mugging in Paris into Kristalnacht.