Which API can I use for tracking the position of animal in one or more images? I'd like to build an application for tracking the position of a given animal (e.g. a cat) in a series of images.

Is there any off-the-shelf API I could use?

Azure has some Vision APIs, but it seems to me they can't be used to get the position of something in an image.
FastAI is the most “out of the box” API for this type of task.

For video examples (and a little theory) check out the MOOC section of their site. 

Practical Deep Learning and Cutting Edge Deep Learning are the two sections most relevant to you. 

But if you want a working implementation check out this GitHub repo that implements SSD for your purposes. I can’t say how simple the API is but it does what you are seeking (in pure PyTorch).

note: FastAI was originally built on top of PyTorch (although they are expanding out now), so you’ll be using PyTorch and need a rough idea of how to work with tensors. Most of the challenges are already implemented in their api. Installing their library will automatically install PyTorch.