First off, the ANY will not work properly as you want the customer to be able to search for any 1 work that matches, so if I search for 'Aspirin Capsules' and only Aspirin exists, it should still return a match, now to do that in access you may have to split the search at every space and loop through to construct your sql statement.
To do a Begins with you simply do a
& "Drugname LIKE '" & Request("Search") & "%'"
just to leave out the % in the beginning....
Another note, don't use Request use Request.Query or .Form to limit vulnerabilities.