Two ways to do this.
You can show the reimbursements you get from the customer as General Income, on the Sch C business and then deduct 100% of the travel expenses.
Another way is to total 100% of the travel expenses, and then reduce the amount entered on the return by the reimbursement.
Whatever method you use, document it in the tax file and be consistent year to year. This will help in comparing years to each other and also not to overlook this.
Either method is acceptable to IRS.
I prefer the reduce the travel expenses by the amount reimbursed.