If the evar is set to persist beyond Hit (preferably Visit) and the allocation setting is chosen to "Most Recent", the evar can be broken by Product variable. That report should tell us which internal search term was used to finally purchase a product.
Yes, it is possible to find which product the visitor has purchased after searching for a specific keyword. My suggestion will be to use a merchandising eVar with the allocation as "most recent (last)" & expiration after purchase. For more information on the merchandising eVars you can refer to the below documentation link.