In the sentence "You can search site by site," what type of phrase is "site by site"? And am I correct in thinking that it does not need hyphens in that particular sentence?
You can search quickly.
Here, quickly is an adverb that modifies search.
You can search site by site.
In the same way, site by site is an adverbial phrase that modifies search. (It describes how you will search.)
Compound adjectives used in front of nouns are commonly (although not always) hyphenated:
It is a site-by-site search.
However, when a compound adjective comes after a noun, the hyphenation is normally dropped:
The search is site by site.
In your example sentence, site by site not only comes afterwards, but it's being used adverbially rather and adjectivally.