Is it possible to install an external GPU in a Dell E6400 [EGPU]

No eGPU for this model is going to offer much performance -- and you'll wind up with a notebook that becomes a desktop anyway (you won't be able to close up the system after you install and wire, plus you'll need an external monitor).

Take whatever the eGPU box costs and buy a decent, perhaps second hand desktop system -- that's what you'll end up with anyway, without the performance bottleneck of a notebook PCIe interface.