It depends on what style guide you ascribe to and what your needs as a writer are. I'll run through a few of the variants to show the range of options.
Pragmatic: Chicago Manual of Style
Since you mention the Chicago Manual of Style, the manual's 17th edition has an entire subsection (7.81-7.89) on when to hyphenate words. The manual recommends that usage should depend on the situation and the writer's needs. Here are a few useful guidelines for hyphenating a suffix like -goer:
7.81 suggests looking in the dictionary for any examples documented there.
7.83 suggests that, over time, hyphenated compounds tend to become closed, though this tendency isn't universal.
7.84 suggests that the hyphen can be used to avoid mispronunciation or misreading:
A hyphen can make for easier reading by showing structure and, often, pronunciation. Words that might otherwise be misread, such as re-creation [compare recreation] or co-op [compare coop], should be hyphenated
So Chicago essentially makes a series of suggestions and gives the toolbox to its writers to determine when to use a hyphen with goer.
Absolutely No Hyphen: AP Style
If you prefer an absolute answer, AP has issued its recommendation in a tweet:
AP Style tip: No hyphen when adding -goer to a word: concertgoer, moviegoer, partygoer, theatergoer.
AP tends toward absolutes because the work of the Associated Press demands consistency. They would not put a hyphen.
Old School: A Dictionary of English Usage
If you want an old-school take on hyphens, the 2009 edition of Fowler's dictionary edited by David Crystal should prove useful. Fowler was dour about hyphens, starting his entry on them with the following:
The chaos prevailing among writers or printers or both regarding the use of hyphens is discreditable to English education.
Eventually Fowler sets down a series of rules. His first rule gives an instructive example:
A hyphen is a symbol conveying that two or more words are made into one ; the union may be for the occasion only (as in most of the examples above), or permanent (as in fire-irons, committee-man)
Committee-man is similar to movie-goer, as the second element in both uses refers to a person. That said, Fowler's second rule suggests that, over time, such a hyphen should tend to disappear unless mitigating circumstances arise to require its use:
The hyphen is not an ornament; it should never be placed between two words that do not require uniting & can do their work equally well separate ; & on the other hand the conversion of a hyphened word into an unhyphened single one is desirable as soon as the novelty of the combination has worn off, if there are no obstacles in the way of awkward spelling, obscurity, or the like.
In other words, even Fowler's advice admits that, decades later, either movie-goer or moviegoer could make sense, depending on whether the hyphen were seen as permanent and whether the novelty of the combination had worn off.