It might be that the connection opened using the Jet provider can't handle a recordset and an update query at the same time. Just use an implicit connection like this:
Code:
Set MyConn = Server.CreateObject("ADODB.Connection")
Set RS = Server.CreateObject("ADODB.Recordset")
connectString = "Driver={Microsoft Access Driver (*.mdb)}; DBQ=" & MdbFilePath & ";"
MdbFilePath = Server.MapPath("directory\todatabasefile.mdb")
'open the recordset
SQL_query = "SELECT * FROM files"
RS.Open SQL_query, connectString
'run the update query
SQL_query = "UPDATE myTable SET myField = 'foo'"
MyConn.Open connectString
MyConn.Execute SQL_Query
If you're running SQL you should ALWAYS use parameterized queries or stored procedures. SQL injection is the easiest way for people to completely screw up your database.