The standard MathJax TeX font does not include glyphs for lowercase script letters, only uppercase ones. If you are managing your own website where you want this character, you can configure MathJax to use the STIX fonts in either the HTML-CSS or SVG output renderers, but not CommonHTML.
Alternatively, you could reference the Math Alphabet positions for these characters, U+1D4C7 for script r, and U+1D4FB for bold script r. So something like
\def\scriptr{\unicode[.43,0][STIXGeneral, Cambria Math]{x1D4C7}}
\vec\scriptr \equiv \vec r - \smash{\vec r}'
in a web page should do it. Note, however, that this relies on the user having a font that contains this character, and not many do. I've listed STIXGeneral and Cambria Math here, as both of those do, but there is no guarantee that the user will hav such a font available.
Davide