There are a number of different financing methods you can use to acquire an existing business.
Vendor take-back financing: Sometimes, the owner himself can provide you with the financing you need to buy his business.
Secured business loan: Often times when you want to purchase an existing business, there will be equipment that comes with it. If the equipment still has enough value, it can be used as collateral to secure a loan.
Bank loan: The most traditional way you can finance your purchase is by seeking a loan from the bank. Though it may be hard to get approved, it is an option.