Fees for each of Budget's airline partners can be found here:
https://www.budget.com/en/programs/m...tners/airlines
For AAdvantage, it's $1.00/day, but that fee will be subject to tax in some locations.
Someone posted on FT a while back about earning miles at an off-airport Avis or Budget location in Las Vegas, NV, but that seems to be very much the exception.