Can I get Jenkins to build a git tag from a passed in parameter?

There's Git Parameter Plugin, which allows you to do exactly that:

This plugin allows you to assign git tag or revision number as parameter in Parametrized builds. There is no need to set up anything special, this plugin will read your default configuration from Git Plugin.

Will up oooold topic, since this one is in google's top. Spent some time on this question... Short answer: Extensible choice plugin + groovy script. This allows to make dropdown menu already filled with existing tags.

def gettags = "git ls-remote -t [email protected]:mycompany/com.someproject.git".execute()
def tags = []
def t1 = []
gettags.text.eachLine {tags.add(it)}
for(i in tags)
    t1.add(i.split()[1].replaceAll('\\^\\{\\}', '').replaceAll('refs/tags/', ''))
t1 = t1.unique()
return t1

Long answer here

Make the build parameterized and in the git URL box, put the name of the variable you've defined. For example: ${GIT_URL}. This should do it.