Is there a Unicode character for the share icon?

Short answer: no.

Long answer: https://en.wikipedia.org/wiki/List_of_Unicode_characters

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As a minor aside / workaround for a bit of fun:

DEMO: http://jsfiddle.net/8rzgv/

<span class="share">&lt;</span>

.share {
    font-weight: bold;
    position: relative;
}
.share:before {
    content: ".";
    font-size: 2.2em;
    position: absolute;
    bottom: -2.5px;
    left: -4px;
}
.share:after {
    content: ":";
    font-size: 2em;
    position: absolute;
    bottom: -6px;
    right: -7px;
}

Since that share symbol doesn't exist on unicode range u0000-uFFFF, you may use instead an iconic font and import that symbol. E.g., if you look at http://fontello.com/ and you search for "share" you will find a similar symbol under Entypo font

(with Entypo the unicode sequence is \uE800). These fonts are also in SVG format and the size/color/tickness of the symbols can be managed by CSS properties