Fastest Gun in the West

Snails, 71 bytes

Try it online.

A
\/a=\)2w\V={\\u\(}c=\/b={\\c\(2}u!{(nb|.|.n.)=^o=^O=^0^ }{nb|.|.n.}^

Mathematica, 170 bytes

Catch@BlockMap[b="O"|"o"|"0";MatchQ[#,p=Characters@{"  (X)  ","(X\\ /X)"," (XVX) ","  \\X/  "}/." "->_/."X"->$;p/.$->b|"X"|" "]&&Throw@Count[Pick[#,p,$],b,2]&,#,{4,7},1]&

Takes an array of strings/chars. Returns the number of bullets.


JavaScript, 215 211 209 bytes

Thanks to Shaggy for -4 bytes!

f=

i=>[...i].map((_,x)=>eval(String.raw`a=i.match(/(\n|^).{${x}}..\(${Z='([oO0 ])'}\).*\n.{${x}}\(${Z}\\.\/${Z}\).*\n.{${x}}.\(${Z}V${Z}\).*\n.{${x}}..\\${Z}\//);a&&alert(a.filter(i=>/^[oO0]$/.test(i)).length)`))

tests = [["  (0)   ( )\n(o\\ /o( \\ / )\n (oVo) ( V )\n  \\o/   \\ /",0],["----------------------------\n////////////////////////////\n////////////////////////////\n/////////////(o)////////////\n///////////(0\\// )//////////\n////////////( Vo)///////////\n/////////////\\ /////////////\n////////////////////////////\n----------------------------",3],["()()()()()()()()()()()()()()()()\\)/)()()()()()()()()()()()()\n()()()()()()()()()()()()()()()()(V)()()()()()()()()()()()()(\n()()()()()()()(\\0/)()()()()()()()()()()()()()()()()()()()()(\n()()()()()()()()()()()()()()()()()()()()()()()( )()()()()()(\n()()()()()()(o)()()()()()()()(00)()()()()()(( \\(/0)()()()()(\n()()()()()()()()()()()()()()()()()()()()()()(( V )()()()()()\n()()()()()()()()()()()()()()()()()()()()()()()\\O/()()()()()(",2],["------(0)---\n||||(0\\|/0)\n-----(0V0)--\n||||||\\0/\n-------_------\n|||||||-|\n------|-|\n|||||||-|\n------|-|",6],["00ooOO(0)/\\\\/V\n\n  ( )\n( \\\\/ )\n ( V )\n \\\\ /\n\n00OOoo()()()())))",0],["  (\\)     (0)\n(0\\\\/0) ( \\0/ )\n (0V0)   ( V )\n \\\\0/     \\ /",1]]

alert = x => i = x;

tests.forEach((t, index) => {
    i = 'Not Found'
    f(t[0]);
    document.getElementById('a').textContent += t[0] + '\n\n' + i + '\n\n';
    if (i != t[1]) {
        document.getElementById('a').textContent += 'Failed test ' + index + '! Expected '  + t[1] + '\n\n';
        document.getElementById('a').style = 'color: red'
    }
})
<pre id="a"></pre>

Basically, tries to match a gun n characters after a line break, for n from 0 to the length of the string.

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String

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