Welcome to the forums
Just to rephrase your question:
There is a site you visit which is not under your control.
You use the site's search functionality a lot.
You are tired of manually selecting "Exact Match" from the search drop down every time you search.
You are therefore looking for a way for this option to be pre-selected as soon as you visit the site.
Is this correct?
If so, could you provide a link to the site in question?