There are more than one way to do this.  Here is a simple solution:

Dim strSQL as String
Dim dbf as Database
Dim tdfs as TableDefs
Dim tdf as TableDef

'Set up object variables
   Set dbf = CurrentDb
   Set tdfs = dbf.TableDefs
   Set tdf = tdfs("MyQueryName")

'Get the SQL string for the query
   strSQL = tdf.SQL
'Replace the old name with the new name in the SQL
   strSQL = Replace(strSQL, "OldTableName", "NewTableName")

'Runs the append query with the selected table name
   dbf.Execute(strSQL)

'Note, we don't change the name in the original saved query so we don't know
what it is next time we need it.  The Execute method above is the same as
running the query any other way and is about 5 times faster than then RunSql
method.

'Destroy object variables
   Set tdf = Nothing
   Set tdfs = Nothing
   Set dbf = Nothing

On my website (www.rogersaccesslibrary.com), is a small Access database
sample called "CreateQueries2.mdb" which illustrates how to do this.  All
the queries in the samples are Select queries, but the process is the same.
The SQL is just different.