If the user knows the table name then test: 
Static TName As String
TName = InputBox("Enter table name", "Access Table name", TName)
Application.DisplayAlerts = False
With ActiveSheet.ListObjects.Add(SourceType:=0, Source:=Array("OLEDB;Provider=Microsoft.ACE.OLEDB.12.0;Password="""";User ID=Admin;Data Source=\\Prdhilfs03\l&i-sales&mkt\WORKAREA\Agencyservices\Accou", "nting\AGY\Databases\DatabaseAudit.mdb;Mode=Share Deny Write;Extended Properties="""";Jet OLEDB:System database="""";Jet OLEDB:", "Registry Path="""";Jet OLEDB: Database Password="""";Jet OLEDB:Engine Type=5;Jet OLEDB: Database Locking Mode=0;Jet OLEDB:Global Parti", "al Bulk Ops=2;Jet OLEDB:Global Bulk Transactions=1;Jet OLEDB:New Database Password="""";Jet OLEDB:Create System Database=False;Jet", " OLEDB:Encrypt Database=False;Jet OLEDB: Don't Copy Locale on Compact=False;Jet OLEDB:Compact Without Replica Repair=False;Jet OL", "EDB:SFP=False;Jet OLEDB:Support Complex Data=False;Jet OLEDB:Bypass UserInfo Validation=False"), Destination:=Range("$A$1")).QueryTable
  .CommandType = xlCmdTable
  .CommandText = TName
  .RowNumbers = False
  .FillAdjacentFormulas = False
  .PreserveFormatting = False
  .RefreshOnFileOpen = False
  .BackgroundQuery = True
  .RefreshStyle = xlInsertDeleteCells
  .SavePassword = False
  .SaveData = True
  .AdjustColumnWidth = True
  .RefreshPeriod = 0
  .PreserveColumnInfo = True
  .SourceDataFile = "\\Prdhilfs03\l&i-sales&mkt\WORKAREA\services\Accounting\AGY\Databases\DatabaseAudit.mdb"
  .ListObject.DisplayName = "Table_DatabaseAudit"
  .Refresh BackgroundQuery:=False
End With
Application.DisplayAlerts = True
Additional/Changed code in red:

Static TName As String 'conventionally placed at top of sub.
TName = InputBox("Enter table name", "Access Table name", TName)
<snip>
.CommandType = xlCmdTable
.CommandText = TName
.RowNumbers = False
<snip>

If however, the user doesn't know what the table name might be, do you?

ps. Quite surprised by brevity of that .CommandText line, although .CommandType = xlCmdTable tends to explain it!