Is there Unicode glyph Symbol to represent "Search"

I'd recommend using http://shapecatcher.com/ to help search for unicode characters. It allows you to draw the shape you're after, and then lists the closest matches to that shape.

There is U+1F50D LEFT-POINTING MAGNIFYING GLASS (🔍) and U+1F50E RIGHT-POINTING MAGNIFYING GLASS (🔎).

You should use (in HTML) 🔍 or 🔎

They are, however not supported by many fonts (fileformat.info only lists a few fonts as supporting the Codepoint with a proper glyph).

Also note that they are outside of the BMP, so some Unicode-capable software might have problems rendering them, even if they have fonts that support them.

Generally Unicode Glyphs can be searched using a site such as fileformat.info. This searches "only" in the names and properties of the Unicode glyphs, but they usually contain enough metadata to allow for good search results (for this answer I searched for "glass" and browsed the resulting list, for example).

Note that you can use Unicode Variant Selectors to explicitly pick how the glyphs will be rendered. Specifically VS15 (U+FE0E) for text-style and VS16 (U+FE0F) for emoji-style are relevant here. Simply append them to your chosen Unicode symbol to indicated if you'd like it to be rendered as text or as an emoji (assuming the software/browser supports the selectors and the relevant representation):

Note that the result might vary depending on platform. For example Chrome seems to always render this character as an Emoji on both Linux and Windows.

Displayed correct at Chrome OS - screenshots from this system.

༗ U+0F17

⌕ U+2315

ᰄ U+1C04

Use the ⚲ symbol (encoded as ⚲ or ⚲), and rotate it to achieve the desired effect:

<div style="-webkit-transform: rotate(45deg); 
               -moz-transform: rotate(45deg); 
                 -o-transform: rotate(45deg);
                    transform: rotate(45deg);">
    &#9906;
</div>

It rotates a symbol :)