Suggestions:

1. Add the # delimiter to the date.

2. Omit the query name from the WhereCondition string.

3. If you need to include the table name, use square brackets around the
table name and the field name separately, i.e.
       [Distributors].[DistID]

4. The subform is not open in its own right, i.e. it will not be found in
the Forms collection. If it is a subform of the form where you are writing
this code, you can refer to it as:
   Me.[NameOfYourSubformControlHere].Form![NameOfYourControlHere]
If not, use:
   Forms![NameOfYourMainFormHere]
in place of the Me.

5. Create a separate string, so you can Debug.Print the results, and check
them in the Immediate Window after it runs. (Press Ctrl+G.)

6. Brackets are optional, but can help.

7. Use IsNull() to check if there is a value in the 2 controls.

8. Save any edits in progress before you open the report.

The result will be something like this (without the null tests):
   Dim strWhere As String
   If Me.Dirty Then Me.Dirty = False
   strWhere => "([DistID] = " & Me![DistID] & ") AND ([DatePaid] = " & _
       Format(Me.[frm-DistDetailSubform].Form![DatePaid],
"\#mm\/dd\/yyyy\#") & ")"
   Debug.Print strWhere
   DoCmd.OpenReport stDocName, acViewPreview, , strWhere