In a conversation about a software system between people familiar with the source code, it is useful to use method calls as verbs, even when their normal definition does not match the meaning in this particular context. For example, "the object does not validate" instead of "the object's validate method returns false".
When talking to a wider audience (i.e. anyone not familiar with the source code) it is advisable to avoid jargon as far as possible, and use terms and phrases that are widely understood. When considering acceptability to a wider audience, we can look at prescriptive language rules, and observed general usage patterns.
From a prescriptive perspective, when talking about clicking an item in an electronic interface, click should normally be used as a transitive verb, i.e. "the user clicks an item", or "the user clicks on an item".
However, when used as an intransitive verb, click has meanings that could make sense in your scenario. If the interface is expected to give audible feedback when selecting an item, then it would be perfectly acceptable to complain when it "does not click".
Other meanings of click as an intransitive verb include "function smoothly", or "succeed", but these would perhaps be better applied to overall functioning of a system, rather than a single action within it.
It is certainly possible that your phrase may gain acceptance as a way of describing the failure of an item in a user interface to respond to being clicked on. In fact extending a transitive verb with an intransitive usage seems to be popular. For example, while "validate" is normally a transitive verb, it is very common for people to use it intransitively (e.g. "My ticket won't validate.").
Given the above, I would surmise that this usage of "It won't click" could be judged an acceptable colloquialism. Personally, I would prefer to say "It does not respond when I click on it."
Sources: Merriam-Webster definitions of "click" and "validate".
Tangentially, unless debugging or logging, saying the event triggered by the OnClick handler is not happening is guessing the fault from the symptom. Maybe there is no event registered. Maybe the handler is not being called.