Sass code for displaying two simple search forms
Sass

If this is meant to be a reusable Sass library, you'll want to make use of default variables so that they're easier to override:

$container-width: 940px !default;
$search-container-padding: 54px !default;


You'll want to do something similar with your mixins so that the values can be changed for one-off instances:

@mixin input-width($width, $form-width: $default-form-width, $button-width: $default-button-width, $input-inset: $default-input-inset)
{
    .search-input
    {
        width:$width;
    }
    .search-placeholder
    {
        left:$form-width - $width - $button-width + $input-inset;
    }
}


You'll probably want to avoid generating styles by default, since the user might not want them.  It is a common request for people who use libraries like Bootstrap and Foundation:  how do I make it stop generating default styles?  Instead, you might want to consider having a separate file that can be imported that generates these styles.

The $input-br variable has an ambiguous name.  When I initially saw it in the source, I thought it was for the br element.  It wasn't until later I realized br was being used as an abbreviation for border-radius (none of your other variables are abbreviated).

Sass has a bit of a shortcut you can use for some properties:

border: {
  top: $input-border;
  bottom:$input-border;
  left:$input-border;
  radius: $input-br 0 0 $input-br;
}


CSS

If the background image for your button fails to load, the button is completely invisible because it is white text on a white background.

The use of px units here is sub-optimal.  When your containers contain text, you should be using a unit that's relative to the font-size, like em or ex, to specify its measurements.  If the user changes their browser's font settings in such a way that they want their font size to be 20px at the bare minimum, then your content is going to spill out of these containers.

Using archaic widths of yesteryear (900px~1024px) are very unfriendly towards mobile devices.  Set a max-width if you must, but avoid setting hard widths that are likely to cause horizontal scrolling.  Better yet, look into responsive design.

HTML

Placeholder text is not meant to be a replacement for labels, please don't treat it as such.  It should be thought of as example content only.

Labels must have a closing tag.  If they're not containing the input element that they're for, they need to have a for attribute and the associated input element needs to have an id with the matching value.

There's a new input type of search that I recommend using instead of text for your search input fields.  Browsers that don't support search will fallback and use text anyway.

Your 2nd submit button has a class of submit rather than a type.