Depends on the site.
No, you can't actually fill in the field in the <form> on the site.
But if the site allows it, you could have a <form> *IN THE EMAIL HTML* that duplicates the <form> on the site, allow the user to click on its submit button, and submit the <form> to keepvid.com's search page.
Many sites won't allow this: They insist that <form>s be posted only from within the same site (and often even from a specific page on the site).
There are sometimes ways around this, by "spoofing" the target site, but they involve server-side coding (that is, the email link would actually go to a server-side page on your site which would spoof keepvid.com). And truly aggressive sites (e.g., google.com) have multiple ways to prevent even this from working.
An optimist sees the glass as half full.
A pessimist sees the glass as half empty.
A realist drinks it no matter how much there is.