Without Excel Tables
This is a bit of an odd way to do it but if you have a lot of cells to do, it's possibly faster than copy/paste special:
ThisWorkbook.Worksheets("Archive").Range("A2:D2").Formula = "=INDEX('Material Check'!$B$3:$B$6,Column())"
ThisWorkbook.Worksheets("Archive").Range("A2:D2").Value = ThisWorkbook.Worksheets("Archive").Range("A2:D2").Value
The first line populates the destination range with a formula that pulls the data from the source, using INDEX/COLUMN to transpose the result.
The second line simply converts the formula to hard values.
EDIT - Solution to copy the values to the bottom of the list
Using Excel Table
To do this you will need to go to "Insert" --> "Table".
''Get a reference to your destination table
Dim Tbl1 As ListObject
Set Tbl1 = ThisWorkbook.Sheets("Archive").ListObjects("Table1") ''Change these to your destination sheet/table names
''add a new row to the table
Dim Newrow As ListRow
Set Newrow = Tbl1.ListRows.Add
''populate the new row
With Newrow
.Range(Tbl1.ListColumns("Column1").Index) = ThisWorkbook.Worksheets("Material Check").Range("B3") ''change these to your destination column name and your source sheet/ranges
.Range(Tbl1.ListColumns("Column2").Index) = ThisWorkbook.Worksheets("Material Check").Range("B4")
.Range(Tbl1.ListColumns("Column3").Index) = ThisWorkbook.Worksheets("Material Check").Range("B5")
.Range(Tbl1.ListColumns("Column4").Index) = ThisWorkbook.Worksheets("Material Check").Range("B6")
End With