Update 2021-07-21

It's been more than half a decade since I first wrote this answer. The extensions to which I originally linked are abandoned, and Visual Studio Code's intrinsic PHP support hasn't improved, which is disappointing. The only decent extension still standing of which I'm aware is PHP Intelephense, which uses a freemium model: basic features are free, and a lifetime license is $12 USD as of writing.

The free version of Intelephense supports code formatting with the usual shortcuts (Alt + Shift + F on Windows and Linux, ⌥⇧F on macOS). Visual Studio Code continues to lack built-in support for PHP code formatting and will direct you to the extension marketplace if you attempt to format PHP without an appropriate extension installed.

Original answer

Visual Studio Code has pretty awesome PHP support. What it lacks is covered by extensions. A quick search reveals at least three (1, 2, and 3) that claim to support PHP formatting.

They mostly seem to use the standard shortcut of Alt + Shift + F on Windows/Linux, with varying shortcuts on Mac. If you're on Mac, give ⌥⇧F a try.

Well here is my very basic and rough script:

#!/usr/bin/php
<?php
class Token {
    public $type;
    public $contents;

    public function __construct($rawToken) {
        if (is_array($rawToken)) {
            $this->type = $rawToken[0];
            $this->contents = $rawToken[1];
        } else {
            $this->type = -1;
            $this->contents = $rawToken;
        }
    }
}

$file = $argv[1];
$code = file_get_contents($file);

$rawTokens = token_get_all($code);
$tokens = array();
foreach ($rawTokens as $rawToken) {
    $tokens[] = new Token($rawToken);
}

function skipWhitespace(&$tokens, &$i) {
    global $lineNo;
    $i++;
    $token = $tokens[$i];
    while ($token->type == T_WHITESPACE) {
        $lineNo += substr($token->contents, "\n");
        $i++;
        $token = $tokens[$i];
    }
}

function nextToken(&$j) {
    global $tokens, $i;
    $j = $i;
    do {
        $j++;
        $token = $tokens[$j];
    } while ($token->type == T_WHITESPACE);
    return $token;
}

$OPERATORS = array('=', '.', '+', '-', '*', '/', '%', '||', '&&', '+=', '-=', '*=', '/=', '.=', '%=', '==', '!=', '<=', '>=', '<', '>', '===', '!==');

$IMPORT_STATEMENTS = array(T_REQUIRE, T_REQUIRE_ONCE, T_INCLUDE, T_INCLUDE_ONCE);

$CONTROL_STRUCTURES = array(T_IF, T_ELSEIF, T_FOREACH, T_FOR, T_WHILE, T_SWITCH, T_ELSE);
$WHITESPACE_BEFORE = array('?', '{', '=>');
$WHITESPACE_AFTER = array(',', '?', '=>');

foreach ($OPERATORS as $op) {
    $WHITESPACE_BEFORE[] = $op;
    $WHITESPACE_AFTER[] = $op;
}

$matchingTernary = false;

// First pass - filter out unwanted tokens
$filteredTokens = array();
for ($i = 0, $n = count($tokens); $i < $n; $i++) {
    $token = $tokens[$i];
    if ($token->contents == '?') {
        $matchingTernary = true;
    }
    if (in_array($token->type, $IMPORT_STATEMENTS) && nextToken($j)->contents == '(') {
        $filteredTokens[] = $token;
        if ($tokens[$i + 1]->type != T_WHITESPACE) {
            $filteredTokens[] = new Token(array(T_WHITESPACE, ' '));
        }
        $i = $j;
        do {
            $i++;
            $token = $tokens[$i];
            if ($token->contents != ')') {
                $filteredTokens[] = $token;
            }
        } while ($token->contents != ')');
    } elseif ($token->type == T_ELSE && nextToken($j)->type == T_IF) {
        $i = $j;
        $filteredTokens[] = new Token(array(T_ELSEIF, 'elseif'));
    } elseif ($token->contents == ':') {
        if ($matchingTernary) {
            $matchingTernary = false;
        } elseif ($tokens[$i - 1]->type == T_WHITESPACE) {
            array_pop($filteredTokens); // Remove whitespace before
        }
        $filteredTokens[] = $token;
    } else {
        $filteredTokens[] = $token;
    }
}
$tokens = $filteredTokens;

function isAssocArrayVariable($offset = 0) {
    global $tokens, $i;
    $j = $i + $offset;
    return $tokens[$j]->type == T_VARIABLE &&
        $tokens[$j + 1]->contents == '[' &&
        $tokens[$j + 2]->type == T_STRING &&
        preg_match('/[a-z_]+/', $tokens[$j + 2]->contents) &&
        $tokens[$j + 3]->contents == ']';
}

// Second pass - add whitespace
$matchingTernary = false;
$doubleQuote = false;
for ($i = 0, $n = count($tokens); $i < $n; $i++) {
    $token = $tokens[$i];
    if ($token->contents == '?') {
        $matchingTernary = true;
    }
    if ($token->contents == '"' && isAssocArrayVariable(1) && $tokens[$i + 5]->contents == '"') {
        /*
         * Handle case where the only thing quoted is the assoc array variable.
         * Eg. "$value[key]"
         */
        $quote = $tokens[$i++]->contents;
        $var = $tokens[$i++]->contents;
        $openSquareBracket = $tokens[$i++]->contents;
        $str = $tokens[$i++]->contents;
        $closeSquareBracket = $tokens[$i++]->contents;
        $quote = $tokens[$i]->contents;        
        echo $var . "['" . $str . "']";
        $doubleQuote = false;
        continue;
    }
    if ($token->contents == '"') {
        $doubleQuote = !$doubleQuote;
    }
    if ($doubleQuote && $token->contents == '"' && isAssocArrayVariable(1)) {
        // don't echo "
    } elseif ($doubleQuote && isAssocArrayVariable()) {
        if ($tokens[$i - 1]->contents != '"') {
            echo '" . ';
        }
        $var = $token->contents;
        $openSquareBracket = $tokens[++$i]->contents;
        $str = $tokens[++$i]->contents;
        $closeSquareBracket = $tokens[++$i]->contents;
        echo $var . "['" . $str . "']";
        if ($tokens[$i + 1]->contents != '"') {
            echo ' . "';
        } else {
            $i++; // process "
            $doubleQuote = false;
        }
    } elseif ($token->type == T_STRING && $tokens[$i - 1]->contents == '[' && $tokens[$i + 1]->contents == ']') {
        if (preg_match('/[a-z_]+/', $token->contents)) {
            echo "'" . $token->contents . "'";
        } else {
            echo $token->contents;
        }
    } elseif ($token->type == T_ENCAPSED_AND_WHITESPACE || $token->type == T_STRING) {
        echo $token->contents;
    } elseif ($token->contents == '-' && in_array($tokens[$i + 1]->type, array(T_LNUMBER, T_DNUMBER))) {
        echo '-';
    } elseif (in_array($token->type, $CONTROL_STRUCTURES)) {
        echo $token->contents;
        if ($tokens[$i + 1]->type != T_WHITESPACE) {
            echo ' ';
        }
    } elseif ($token->contents == '}' && in_array($tokens[$i + 1]->type, $CONTROL_STRUCTURES)) {
        echo '} ';
    } elseif ($token->contents == '=' && $tokens[$i + 1]->contents == '&') {
        if ($tokens[$i - 1]->type != T_WHITESPACE) {
            echo ' ';
        }
        $i++; // match &
        echo '=&';
        if ($tokens[$i + 1]->type != T_WHITESPACE) {
            echo ' ';          
        }
    } elseif ($token->contents == ':' && $matchingTernary) {
        $matchingTernary = false;
        if ($tokens[$i - 1]->type != T_WHITESPACE) {
            echo ' ';
        }
        echo ':';
        if ($tokens[$i + 1]->type != T_WHITESPACE) {
            echo ' ';
        }
    } elseif (in_array($token->contents, $WHITESPACE_BEFORE) && $tokens[$i - 1]->type != T_WHITESPACE &&
        in_array($token->contents, $WHITESPACE_AFTER) && $tokens[$i + 1]->type != T_WHITESPACE) {
        echo ' ' . $token->contents . ' ';
    } elseif (in_array($token->contents, $WHITESPACE_BEFORE) && $tokens[$i - 1]->type != T_WHITESPACE) {
        echo ' ' . $token->contents;
    } elseif (in_array($token->contents, $WHITESPACE_AFTER) && $tokens[$i + 1]->type != T_WHITESPACE) {
        echo $token->contents . ' ';
    } else {
        echo $token->contents;
    }
}