I have a small recruiting and staffing business. How do I get big companies to do business with me?
Not to sound too flippant, but you have to earn the reputation to do business with the equivalent of an Apple or a Microsoft.
It's like a ladder you have to climb one step at a time, or a mountain--in most cases you won't be able to hit the summit in one day.
Make a target list of "big" companies you want to go after. Do some research. Make a couple of calls. What are they looking for in a staffing company? How could you get on their radar? At what point do they consider a smaller vendor worthy of a business transaction?
In the meantime, get clear on your long-term plans. How big do you want to be? What size companies do you want to serve? What do you have to do to get there?
Best case is to do great work so your reputation precedes you--so one day those "big" companies will be calling on you.
All the best.