To use array formulas within VBA code set ReferenceStyle for Application object to xlR1C1 (temporarily, only during your function is executed). Finally call Evaluate to get the result of the formula.
Private Const TEMPLATE As String = "=INDEX({0},MATCH(1,({1}={2})*({3}={4}),{5}))"
Private Const MATCH_TYPE = 0
Public Function TestIndexMatch1(ByRef outputRange As Range, _
ByRef nameCriteria As Range, _
ByRef dateCriteria As Range, _
ByRef nameRange As Range, _
ByRef dateRange As Range)
On Error GoTo Err_Handler
Err.Number = 0
Dim originalReferenceStyle
originalReferenceStyle = Application.ReferenceStyle
Application.ReferenceStyle = xlR1C1
Dim myFormula As String
myFormula = Replace(TEMPLATE, "{0}", outputRange.Address())
myFormula = Replace(myFormula, "{1}", nameCriteria.Address())
myFormula = Replace(myFormula, "{2}", nameRange.Address())
myFormula = Replace(myFormula, "{3}", dateCriteria.Address())
myFormula = Replace(myFormula, "{4}", dateRange.Address())
myFormula = Replace(myFormula, "{5}", MATCH_TYPE)
TestIndexMatch1 = Application.Evaluate(myFormula)
Err_Handler:
If (Err.Number <> 0) Then MsgBox Err.Description
Application.ReferenceStyle = originalReferenceStyle
End Function
And so it looks on sheet: