Same thing happens with Access when connected to server-side code, such as ASP code.
UPDATEs cannot be seen until the connection is closed and reopened. There are some ways around this restriction, but they are more trouble than they are worth.
And your code is overly complex. You can make it a lot simpler so that closing and reopening the connection is easier.
// treat this as a CONSTANT:
var connString = "Provider=Microsoft.Jet.OLEDB.4.0;Data Source='database.mdb'";
var adoconn = new ActiveXobject("ADODB.Connection");
// there is no need at all for the Command object!!!!
var strSQL = "Select SUM(database_column_name) As Total FROM tablename";
// and doing this is the fastest and easiest way to execute a SELECT into a recordset:
var adoRS = adoconn.Execute(strSQL);
textbox1.value = adoRS.Fields(0);
adoRS.Close(); // don't forget to do this
... now you have some code that updates the DB ...
// so close and reopen the DB
// and re-execute the query
adoRS = adoconn.Execute(strSQL);
I will bet you it works!