Bookkeeping for a logistics company

You should have a company checking account on day one and have enough working capital to make all of these payments. Making payments from a personal account is a complete mess, and tells me that the owner isn't prepared to run a business. I hope I am wrong on that. 

Until then, any amount paid to drivers that comes from the owner's personal account should be totaled and be entered as a deposit to owner equity. 

Any money paid to drivers for things like fuel or repairs should be expensed under each account, with the payment method as cash. Same with subcontractor wages.

But...

As for payments to drivers for services rendered, are they really contractors? There are lots of rules that govern that, and companies can get in big trouble if they fudge on this issue. Make sure that you are choosing status correctly. If they are really employees, you should get a payroll service like Gusto, connect it to your Quickbooks account, and let them do the direct deposits to the workers. 

This is my humble opinion, and your accountant can fine tune your specific situation. 

@agubax 

"The problem is that I have to record a transfer of money from a non-company account (so it is not in QuickBooks) to each driver's personal account which is also not in QuickBooks."

This type of transaction is recorded as a journal entry - debit the appropriate expense account and credit the appropriate owner's equity account.  There is nothing else that needs to be recorded in the company's books.  Is this a sole proprietorship?  LLC?  If so, what is the tax classification of the LLC?  Or, is this an S-corp?  The owner should be aware that paying the drivers from his personal account essentially eliminates the legal protection provided by LLCs and corporations.  He should deposit the money into the business bank account and pay the drivers from that account.