Optimized Algorithm for matching between two (unsorted) data tables?

Oh boy, code with lots of opportunities for simplification. You can reduce the scope of local variables, removing any temptation to assign them unused values. You can also convert For loops to ForEach loops when you don't use the index except to access a collection.

Initial simplification:

private void RunMatchingAlgorithm() {
    foreach (var databaseRow in dtFromTableContracts.Rows) {
        string partNumber2 = databaseRow["supplier_part_no"].ToString();
        string contract_no2 = databaseRow["contract_no"].ToString();
        DateTime expiration_date2 = Convert.ToDateTime(databaseRow["con_end_date"]).Date;

        foreach (var excelRow in dtConversion.Rows) {
            string contract_no = excelRow["vend_contract_no"].ToString();

            //If we have even a partial match, check for a part number match
            if (contract_no2.StartsWith(contract_no)) {
                string partNumber = excelRow["vend_item_id"].ToString();

                //If the values match, populate from both tables
                if (partNumber == partNumber2) {
                    excelRow["wpd_expiration_date"] = expiration_date2.Date;
                    excelRow["wpd_cont_cost"] = databaseRow["contract_cost"];
                    excelRow["wpd_contract_no"] = databaseRow["contract_no"];
                    excelRow["wpd_item_id"] = databaseRow["supplier_part_no"];
                    excelRow["wpd_item_no"] = databaseRow["item_id"];
                    excelRow["discontinued"] = databaseRow["discontinued"];
                    excelRow["job_no"] = databaseRow["job_no"];
                }
            }
        }
    }
}

Come to think of it, this is pretty much the exact case linq queries were designed for. We can transform most of the code into a query:

private void RunMatchingAlgorithm() {
    var matches = from databaseRow in dtFromTableContracts.Rows
                  let partNumber2 = databaseRow["supplier_part_no"].ToString()
                  let contract_no2 = databaseRow["contract_no"].ToString()
                  let expiration_date2 = Convert.ToDateTime(databaseRow["con_end_date"]).Date
                  from excelRow in dtConversion.Rows
                  let contract_no = excelRow["vend_contract_no"].ToString()
                  where contract_no2.StartsWith(contract_no)
                  let partNumber = excelRow["vend_item_id"].ToString()
                  where partNumber == partNumber2
                  select new { databaseRow, excelRow, expiration_date2 }
    foreach (var m in matches) {
        var dst = m.excelRow;
        var src = m.databaseRow;

        dst["wpd_expiration_date"] = m.expiration_date2.Date;
        dst["wpd_cont_cost"] = src["contract_cost"];
        dst["wpd_contract_no"] = src["contract_no"];
        dst["wpd_item_id"] = src["supplier_part_no"];
        dst["wpd_item_no"] = src["item_id"];
        dst["discontinued"] = src["discontinued"];
        dst["job_no"] = src["job_no"];
    }
}

and now I see where an optimization can be applied. We're doing a nested 'from' with a 'where', and that's equivalent to a cross-join. Also, we can cut most of the now-only-used-once temporaries:

private void RunMatchingAlgorithm() {
    var matches = from databaseRow in dtFromTableContracts.Rows
                  join excelRow in dtConversion.Rows
                  on excelRow["vend_item_id"].ToString() equals databaseRow["supplier_part_no"].ToString()
                  where databaseRow["contract_no"].ToString().StartsWith(excelRow["vend_contract_no"].ToString())
                  select new { databaseRow, excelRow }
    foreach (var m in matches) {
        var dst = m.excelRow;
        var src = m.databaseRow;

        dst["wpd_expiration_date"] = Convert.ToDateTime(src["con_end_date"]).Date;
        dst["wpd_cont_cost"] = src["contract_cost"];
        dst["wpd_contract_no"] = src["contract_no"];
        dst["wpd_item_id"] = src["supplier_part_no"];
        dst["wpd_item_no"] = src["item_id"];
        dst["discontinued"] = src["discontinued"];
        dst["job_no"] = src["job_no"];
    }
}

I actually haven't used cross-joins much, but I assume they use a hash table under the hood to have O(n+m) complexity instead of O(n*m). If both tables were in the database, then the database could take advantage of already-constructed hash tables / indexes.

You might also want to consider some sort of generated Linq2SQL class, so you can have type-safe access to row fields.