Can I put the search form at the top of our 404 page?

Yes, no plugins outlets in 404 Page not found, works like a charm with your instructions

paste this one the BODY tag:

<script type="text/javascript">
    var x = document.getElementsByClassName("page-not-found"); 
    var search = '<div class="page-not-found-search-top"><h2>Search this site</h2><p></p><form action="/search" id="discourse-search"><input type="text" name="q" value=""><button class="btn btn-primary">Search</button></form><p></p></div>'
    x.item(0).innerHTML += search
</script>

and paste these in CSS part:

.page-not-found-search {
    display: none;
}

.page-not-found-search-top button {
    margin-left: 10px;
}

.page-not-found-search-top input {
    width: 600px;
}

works in a mobile too…

and thx for your reply

Regards.,