Do I hyphenate between Moose and Viewing in Moose Viewing Capital of Colorado?
Yes, this is a compound modifier preceding a noun, so you should indeed hyphenate "moose-viewing."
Edited to add: Closely analogous examples, all from the New York Times, include the following:
"the hamlet of Roscoe, an unofficial fly-fishing capital" (https://www.nytimes.com/2017/05/25/t-magazine/debruce-hotel-catskills-classic-trash-tee-editors-picks.html?login=email&auth=login-email)
"Hunterdon County has retained its distinction as the deer-hunting capital of New Jersey" (https://www.nytimes.com/1985/01/06/nyregion/the-environment.html)
"Tonight will mark the 34th moon-viewing party in the museum's Japanese garden" (https://www.nytimes.com/1999/08/29/nyregion/the-guide-208795.html)
The phrase could be punctuated (and misinterpreted) in the following three ways:
✘ 1. Moose Viewing Capital of Colorado
A moose that is viewing the capital of Colorado.
✘ 2. Moose-Viewing Capital of Colorado
The capital of Colorado that is viewing moose.
It's possible this could be interpreted in the desired sense—but not necessarily, nor does it follow the normal use of adjectives and nouns.
The road-devouring car.
The soul-sucking job.
In both of these cases, the adjectives are metaphorical, but they still describe things that the nouns do. However, even metaphorically, it makes no sense to say that the capital is viewing moose. In actuality, it's the people living in the capital who are doing the viewing.
✘ 3. Moose Viewing-Capital of Colorado.
The viewing capital of Colorado that moose use.
So, it cannot be left without hyphenation, and neither of the singly hyphenated versions works.
In terms of hyphenation, that leaves this:
✔ Moose-Viewing-Capital of Colorado
Note that in doing this, in order to ensure semantic understanding, it changes the phrase into a compound noun. It becomes something like mother-in-law, where it now acts as a single syntactical unit.
However, even though I believe this is the only technical way to avoid misinterpretations, I think the subtle use of the hyphenation would be lost on most people. Hyphenation (of any form) or not, I think that many people could still find ways to be confused over this phrasing.
I think the best way of avoiding confusion is to rephrase the title.
Moose Capital of Colorado
This describes it as the the best place for moose. What it is about moose is left unstated, but if it's known for moose, the implication would be that it would also be a good place to view them.
Are there different places that are better for different things related to moose, such that one place is the best for viewing them but another is the best for something else? Does the word viewing have to be part of the title?
If so, then one possibility is:
Capital of Moose-Viewing.
Also possible is:
Colorado's Best Place to View Moose
Is capital being used figuratively—or is it talking about Denver, the actual capital? If the latter, then perhaps:
Denver: Colorado's Best Place to View Moose
No. You would only use a hyphen if you were suggesting the capital itself is viewing moose.
According to Grammarbook.com, a hyphen is only used in instances of a compound adjective or verb modifying a noun:
"Generally, hyphenate two or more words when they come before a noun they modify and act as a single idea. This is called a compound adjective.
Examples: an off-campus apartment state-of-the-art design"
Since "moose viewing" is not modifying the word "capital" in this instance, a hyphen is not appropriate.