Co-signing is not the same as co-owning. If you are listed on the title to the car as a co-owner then you can take the vehicle. If you are no a co-owner then you cannot. So see what the title says. And no, you cannot hire a repo company either unless you are a co-owner.
If you are a co-owner then you take the vehicle. If you are not the owner, you can do a three-way chat with the owner, lender and you and see if he will sell the vehicle to you if you want it and can pay for it. Otherwise, if the man does not pay, you have a choice - either allow the car to be repossessed or protect your credit and pay. Once you pay it off, then you can sue the man.
You mention a business. Is the business incorporated or an LLC? Is the truck owned by the business and are you an officer of the business or a member of the LLC? If so, then you could take possession of the vehicle that way if the business owns the truck.
No you cannot do what you propose either. Title is not changed by writing a lender. The vehicle is registered by the DMV. The lender would have possession of the title and they are not going to just put the vehicle in your name because you say so in a letter.
Co-signing for a vehicle for a person is always a bad deal for the co-signer. The fact that this guy needed a co-signer should have been a red flag to you because it meant that the guy either had no credit or bad credit. Its usually the latter (bad credit) which means that he has a bad habit of defaulting on debts and will probably default on the debt that you are being asked to co-sign for before the guy moves on to his next victim.