From the bulbapedia page on Cerulean City: "In the Generation I games and Pokémon FireRed and LeafGreen, the Bike Shop sells Bicycles produced by the company. Designs range from a plain city bike to a fast mountain bike; however, the player can only ever get one: the Bicycle. It can be bought for 1,000,000 or exchanged for a Bike Voucher. As only 999,999 can be held at maximum, there is no way for the player to obtain a Bicycle without a Bike Voucher. The Pokémon Fan Club Chairman in Vermilion City will hand over his Bike Voucher if the player listens to him talk about his Rapidash."
So the answer is No you cannot buy the bike since money is capped at 999,999 which is below the 1 million needed.