I thought it would be a verb considering it is being used in place of "ride in a kayak". However if 'a' is a indefinite article, doesn't it have to refer to a noun? Would that make the word 'kayak' a noun or a verb in that sentence?
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The only verb in the sentence is Go. A kayak here refers to a session of paddling a kayak, as you surmise.
Many verbs have noun forms in the same way. I don't think a kayak is that common, but for example, we say
One could say
which indicates the same activity, but in a continuous state of indefinite time as opposed to a discrete session. For example,
means you kayak once per summer, whereas
means you go at points throughout the summer.
It's a noun - like "go for a swim". But it's not the noun for the concrete object (the type of boat), but rather the act of kayaking.
You seem to be confused about what a noun is. If the word has an article before it, it is, indeed, a noun, regardless of whether it mean a boat or a boat ride. It would only be a verb in something like Kayak along the coast for an hour.
And without context, it is not possible to say which of the meanings is intended in your sentence. The boat ride sense would probably be more common ("I wanted exercise, so I decided to go for a kayak"), but it would be perfectly normal to say "You don't know which boat to choose? Go for a kayak!"