Do I have any hope of rendering with this videocard?

AMD Radeon HD 6310 Graphics. Yes it was built-in and Blender is said to have an issue with Intel built-in GPUs I don’t know if it has issues with AMDs though.

I’m pinned down with CPU rendering and I need GPU support asap.

Drivers I need the right drivers for OpenCL to work.

You can get a nice CUDA-enabled nVidia card for under $100.