Try using the Db database object.  I suspect that using
currentdb().OpenRecordset means that rst and rTarget are both going out of
scope as soon as the line has finished executing.

By the way, it is helpful to tell people the exact error you are getting and
on which line it occurs.  Makes trouble-shooting a lot easier.

dim db as DAO.Database
dim rst as DAO.Recordset
dim rTarget as DAO.Recordset

Set db = CurrentDB()  '<<< -----

'Note the change in the next two lines currentdb to db
Set rst = db.OpenRecordset("QryCustCurrTransactions",dbOpenDynaset)
Set rTarget = db.OpenRecordset("LocalTB", dbOpenDynaset)

Do Until rst.EOF
           rTarget.AddNew
           rTarget!TrnDate = DateSerial(Year(rst!TrnDate),
Month(rst!TrnDate) + 1, 1 - 1)
           rTarget!CustomerID = rst!CustomerID
           rTarget!Debit = rst!DrKD
           rTarget!Credit = rst!CrKD
           rTarget.Update
       rst.MoveNext
   Loop
rst.Close
rTarget.Close

I would think you could do the same thing with an append query and it would
be more efficient

Dim StrSQL

StrSQL = "Insert into LocalTb(TrnDate, CustomerID, Debit, Credit)" & _
   " SELECT DateSerial(Year(TrnDate),Month(TrnDate),0)" & _
   ", CustomerID, DrKD, CrKD" & _
   "  FROM qryCustCurrTransactions"

CurrentDb().Execute StrSQL, DbFailonError

What error number and message are you getting?