Yes, you can deduct mileage as a cash charitable contribution. Charitable miles are at 14 cents per mile for 2021; the rate changes slightly from year to year. You would not be able to include the toll or any other auto related expenses; those are included in the mileage rate.
Also, it does not matter whether you volunteer a full day or not, as long as the mileage was directly connected with your volunteer services and not otherwise reimbursed by the volunteer organization and only incurred by you because of the services you gave.
You should keep contemporaneous records of your volunteer days and note the mileage amount each day. You can then do the math to arrive at a total number of volunteer miles, multiply that by 14 cents, and then add that as a cash charitable contribution to the volunteer organization on Schedule A line 11.
In addition to your commute, if they had you drive your car from the volunteer location to another location (say, to pick up some pet medication or deliver an animal to a client), those miles would also be deductible (again, assuming they don't reimburse you for it).
If you did not itemize, the charitable miles would count towards the charitable contributions of $300/$600 for taxpayers who do not itemize on line 12b of Form 1040.
You're right; it's not common. I have seen some older taxpayers write off medical mileage, which is also uncommon but also perfectly legitimate.
