How can I make my match non greedy in vim?

What's wrong with

%s/style="[^"]*"//g

If you're more comfortable PCRE regex syntax, which

1. supports the non-greedy operator ?, as you asked in OP; and
2. doesn't require backwhacking grouping and cardinality operators (an utterly counterintuitive vim syntax requirement since you're not matching literal characters but specifying operators); and

3. you have [g]vim compiled with perl feature, test using

   :ver and inspect features; if +perl is there you're good to go)

try search/replace using

:perldo s///

Example. Swap src and alt attributes in img tag:

<p class="logo"><a href="/"><img src="/caminoglobal_en/includes/themes/camino/images/header_logo.png" alt=""></a></p>

:perldo s/(src=".*?")\s+(alt=".*?")/$2 $1/

<p class="logo"><a href="/"><img alt="" src="/caminoglobal_en/includes/themes/camino/images/header_logo.png"></a></p>

Instead of .* use .\{-}.

%s/style=".\{-}"//g

Also, see :help non-greedy

Non greedy search in vim is done using {-} operator. Like this:

%s/style=".\{-}"//g

just try:

:help non-greedy