Can a non-profit businees reimburse travel expenses at any rate he chooses? For example, could he repay mileage expense for travel at 15 cents a mile if he chooses? I live in Oregon?

Current standard reimbursement rates are around 37.5 cents per mile. What is the specific policy of the non-profit business in question? Generally, any unreimubursed mileage is tax-deductible. Likewise, underreimbursed mileage is deductible for the difference between the rate you received (e.g. 15 cents) and the standard rate ; see the instructions accompanying your next 1040 form or speak with an accountant who does tax work.