How do I buy a work vehicle so that I can write it off as a business expense?

A receipt is all that is needed. 

You prove that you purchased the vehicle for business by listing it on your tax return.

Most likely you are completing a schedule C, it will either go on section Part IV of the schedule C or on form 4562.

If you are an LLC, then you can have the vehicle titled in the name of the LLC.  You do this with the DMV.

- you need to make sure that your insurance matches the titling.

The vehicle does not need to be in the name of the business or titled under the business name to use it as a business expense. 

Demand that you pay by check otherwise you will not be able to prove the price of the asset.  If he won't do it, buy a different truck. 

You can pay in cash because you will have a Bill of Sale as a receipt.   Make sure it's titled correctly if your business is a separate legal entity and you want your business to own it.

You can't deduct the cost of the vehicle in one tax year.   Vehicles must be depreciated over time.   (Since you said this is a new business, there may be some special rules for start-up costs). 

To deduct the entire cost of vehicle and all mileage or expenses, you'll have to prove not just that you bought it for business but also that you use it exclusively for business.   You will have to choose whether to deduct actual expenses or the standard mileage rate.  You cannot deduct both.  Mileage log showing each use of vehicle, odometer reading at beginning/end and purpose of trip is adequate proof.

If you drive the vehicle for personal and business use, you can only deduct the pro-rated business portion and your mileage log needs to be believable.   If you log in 20 trips to Home Depot or a job location in a month, those miles might not be allowed if you don't have any corresponding Home Depot receipts or client invoices for that job.  Know what I mean?