Find first empty cell in column and enter value from textbox

JB's solution is elegant and I'm certain will be that answer that you need, however the one thing I would mention is that it would enter the value in the next empty cell at the bottom of a column of data (ignoring any empty cells interspersed within the data). If you need to find the first empty cell then your piece of code is the one to utilise:

Private Sub TextBox1_Change()
    Range("A1").End(xldown).Offset(1,0).Value = TextBox1.Value
End Sub

The code returned a value in the columns cells below the selected cell.

For example:
c
co
col
colu
colum
column

Private Sub TextBox1_Change()
Range("A2").End(xlDown).Offset(1, 0).value = TextBox1.value
ActiveCell.value = TextBox1.Text
End Sub