How to run a model in an application using gpu (without CUDA)
I'm pretty sure that you will need CUDA to use the GPU, given you have included the tag tensorflow. All of the ops in tensorflow are written in C++, which the uses the CUDA API to speak to the GPU. Perhaps there are libraries out there for performing matrix multiplication on the GPU without CUDA, but I haven't heard of a deep learning framework that doesn't use CUDA, when executing on the GPU.

What you can perhaps do it find a solution that works using Docker. The latest version doesn't require any additional Nvidia software to be installed. It would mean your customer doesn't need to mess around with CUDA themselves, they just need docker installed, which is generally less than 5 minutes work. Here is the Tensorflow documentation for a starter.

In addition, using Docker is perhaps the most professional and isolated way to deliver code to somebody else's machine. There are many benefits and it works on Windows, Mac, Linux. Any changes you make can simply be pushed to their machine, without them having to change anything. It can also scale arbitrarily to multiple machines, running locally or in the cloud, the list goes on and on...