Here's a longer answer that summarizes my earlier comments:
I'm not aware of any implementations of the ellipsoid algorithm that are practically usable for solving LP's. It's easy enough to cook up an implementation of the ellipsoid method in MATLAB but if you try to solve relatively small test problems (e.g. the afiro test problem from NETLIB which has about 50 constraints and 100 variables) you'll quickly discover the double precision arithmetic isn't up to the task. The problem is that the matrices involved become very ill-conditioned. If you want to play with this, I'd recommend starting with the discussion in Chavatal's Linear Programming textbook (there's an appendix on the ellipsoid method.)
Interior point methods for LP assume that the constraints are given and don't work with a separation oracle, so that's out of the question. However, you might be interested in interior point cutting plane methods that use similar ideas to effectively construct an LP outer relaxation of the feasible set as they go along.
There's a nice set of lecture notes on cutting plane approaches by Boyd and Vandenberghe at
The analytic center cutting plane method (ACCPM) of Goffin and Vial is perhaps the best known method of this type. See the code and references at
Although I'd recommend ACCPM, your problem could also be addressed by a variety of other approaches for general convex optimization problems such as subgradient descent, bundle methods, etc. My expertise is more in interior point methods for LP and conic optimization problems so you might get a better and more up to date answer from someone who works in this area.