If you are a W-2 employee of a service, you can't deduct any work-related expenses on your tax return. If you need someone to pay for children's activities, you need to discuss that with the parents, they should be paying, not you or the service. (I wonder if the parents would give you a pre-paid debit card or a limited use credit card on their account, to pay for activities for the children.)
Separately, the whole point of being reimbursed the standard mileage method is that includes allowances for all vehicle expenses, including gas, maintenance, insurance, and depreciation. For most taxpayers, the standard mileage rate gives you a larger recovery than the actual expense method, and even if it didn't, you can't deduct vehicle expenses (even if a deduction was allowed) after you've been reimbursed using the standard mileage method. That would be getting two tax benefits for one expense (double dipping).
The only thing not covered in the standard mileage method is parking and tolls, and if you have those expenses with the children, you should be talking to the parents.
