Select the best word(s) for the blank in the following sentence.
Each player and coach on staff _________ for a new sports facility by next season.

The word each means that we are talking about a singular subject. Even though there may be more than one player or coach, each of them has this individual hope, so the verb must also be in the singular form, which is hopes.
Also, usually the connecting word, and would create a plural subject while using or would indicate a singular subject. Here, however, the use of each (see above) overrides this and the subject is still singular.
Note that, even though the word hopes appears to be plural, is is the singular form of the verb, to hope. In other words, it is the form of the verb that goes with all singular subjects: he hopes, she hopes etc. A plural subject, like they, would take the plural form of the verb, which has no s: hope (they hope).