If you want to add some code to the search field so that it is only affected in the footer widget area, then you can add this code to your css –
.footer-widgets input[type="search"] {
background: #5a9341;
border: none;
}
If you do it this way then you can add the search field in other places of your site and it will still show up as the default white color. The green will only appear on the search field if used in the footer area. Is this what you’re looking to do?