Here's what I found:

Alamo does not allow their vehicles to be driven into Mexico.

Avis does not allow travel into Mexico.

Budget has restrictions about renting a car to travel into Mexico.... you can only do it from certain locations.

Budget locations in Texas which allow driving into Mexico, include:

Brownsville, Corpus Christi, El Paso, Harlingen, Laredo and McAllen

Vehicles rented from these locations can be taken in the “Free Trade Zone” (approximately 20 miles across the border, and including the Baja Penisula). No specialty vehicles allowed. Mexican insurance must be purchased for. (this averages between $25-$30 per day). Renting a car in Austin to travel into Mexico, would not be available.

Dollar does not allow their vehicles to be driven into Mexico.

Their website lists no general guidelines. They state that you must contact the individual city location from which you wish to rent, in order to determine their policies regarding crossing international borders, including availability and insurance requirements. You would have to contact a local location about renting a car in Austin to travel into Mexico.

Hertz

Their website states that travel into Mexico is only available at specific Hertz locations. They do not allow for policy search on this, specific to renting a car in Austin; the site states that

"The customer must check at the renting location at the time of rental for authorization and purchase of the mandatory Mexican Insurance Policy. The basic Mexican insurance coverage allows travel up to 250 miles into Mexico."