Although Dodge Avenger is a rebadged Mitsubishi Eclipse with Mitsubishi-built 2.5L motor, it uses Chrysler-built trans, and it does have a TCM. You cannot find a TCM from any auto part stores, as it is only available from the dealer or a wrecking yard. You have to make sure you get the same number on the "new" one with your old one. As far as I know, Chrysler made four different TCMs from 95 - 00 model year.
I am looking at a Dodge Stratus, but yours should be about the same. It is a silver box under the hood, in between your air filter box and your upper radiator hose.