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There is no such character as a precomposed character (i.e., as a single encoded character, a code point assigned to a character), but you can in principle represent it using an arrow character followed by a combining overlay character.
The character “⇏” U+21CF RIGHTWARDS DOUBLE ARROW WITH STROKE has been defined as having the canonical decomposition RIGHTWARDS DOUBLE ARROW (U+21D2) COMBINING LONG SOLIDUS OVERLAY (U+0338). In principle, a character should be expected to be rendered the same way as its canonical decomposition. In practice, things don’t always go that way.
Along the same lines, a downwards double arrow with stroke could be written as the two-character sequence DOWNWARDS DOUBLE ARROW (U+21D3) COMBINING LONG SOLIDUS OVERLAY (U+0338) or, in HTML, as ⇓̸. In practice, few fonts contain these characters, and browsers may fail to implement the combination properly. Moreover, in many fonts, the result is awkward. In Arial Unicode MS and in DejaVu Serif, the result might be acceptable, but only the latter is free (can be legally used as a downloadable font via @font-face). Here’s the combination as rendered by your browser with the SO stylesheets in effect: ⇓̸.
It doesn't seem to exist, according to this page (compared to this).