Is there a single word to describe a person who is experiencing something?

There is a generic word, but it is unlikely to be of use to you in anything other than academic writing. This is:

subject

It is quite difficult to find a dictionary definition of this precise use, the nearest being (Collins):

In an experiment or piece of research, the subject is the person or animal that is being tested or studied.

with my iPhone Chambers (not on line) having the similar:

A person or thing on which an operation is performed

This may sound too passive, but a Google ngram search for “subject experienced” gives a plethora of hits of the type:

“The moment-based approach derives the experienced utility of an episode from real-time measures of the pleasure and pain (moment utility) that the subject experienced during that episode.”

Aren’t you glad you did’t write that!

Nobody talks of “experiences” in general other than pseuds and fairground vendors, and even in those contexts you are best using the specific name (‘hippie’, ‘customer’ or the like).

As Jason mentioned in his comment to your question, the word you are looking for is experiencer. Someone that is generally experiencing something.

Experiencer

A person or thing that experiences.

-- Dictionary.com

One that experiences

-- Meriam Webster

This is not a very popular word, because as you suggested, you would usually refer to the act the person is experiencing. For example a trainee is training, a runner is running, they are also an experiencer experiencing the act of training or running of course.

Here is an example in context (although this is speaking of a specific type of experiencer, a near death experiencer):

THINGS TO CONSIDER AS AN EXPERIENCER

...Many experiencers find that PMH Atwater's books help them...Research has shown a number of common after-effects in the experiencer.

-- from International Association for Near Death Studies