Install this extension: https://addons.mozilla.org/en-US/firefox/addon/youtube-opensearch-engine/
Background Info
The underlying standard is OpenSearch.
This is what I downloaded from https://youtube.com/opensearch :
<?xml version="1.0" encoding="UTF-8"?>
<OpenSearchDescription xmlns="http://a9.com/-/spec/opensearch/1.1/">
<ShortName>YouTube</ShortName>
<Description>Search YouTube</Description>
<Tags>youtube video</Tags>
<Image height="16" width="16" type="image/vnd.microsoft.icon">https://www.youtube.com/favicon.ico</Image>
<Url type="text/html" template="https://www.youtube.com/results?search_query={searchTerms}&page={startPage?}&utm_source=opensearch"></Url>
<Query role="example" searchTerms="cat"></Query>
</OpenSearchDescription>
