UserForm animation gets stuck

Your way of animation looks too complicated for me...

Try the next approach, please:

Sub DotAnimation()
   Dim i As Long, frm As Object
   Set frm = LoadingInternet
   If Not IsLoaded(frm.Name) Then Exit Sub
   frm.DotLabelloading.Caption = "."
   For i = 1 To 1000
        DoEvents
        Select Case i
            Case 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000
                frm.DotLabelloading.Caption = frm.DotLabelloading.Caption & "."
        End Select
    Next i
End Sub

and IsLoaded function, modified a little:

Public Function IsLoaded(form As String) As Boolean
 Dim frm As Object
 For Each frm In VBA.UserForms
    If frm.Name = form Then
        If frm.Visible = True Then
            IsLoaded = True: Exit Function
        End If
    End If
 Next frm
End Function

If the above Sub does not work, try please, the next one, for a Button click event (from the form in discussion):

Private Sub CommandButtonX_Click()
 Dim i As Long
 Me.DotLabelloading.Caption = "."
 For i = 1 To 1000
    DoEvents
    Select Case i
        Case 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000
            Me.DotLabelloading.Caption = Me.DotLabelloading.Caption & "."
    End Select
 Next i
End Sub

In order to have a continuous animation loop you can use a recursive Sub like the following one. Of course, you can extend the number of dots per cycle or the dots apparition speed. You can also use an (API) Timer. I've just playing with iteration to check if the animation works:

Private Sub DotAnimation()
 Dim i As Long
 Static AnimNo As Long
 Me.DotLabelloading.Caption = "."
 For i = 1 To 1000
    DoEvents
    Select Case i
        Case 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000
            Me.DotLabelloading.Caption = Me.DotLabelloading.Caption & "."
    End Select
 Next i
 AnimNo = AnimNo + 1
 If AnimNo <= 4 Then
   DotAnimation
 Else
   AnimNo = 0
 End If
End Sub

And, instead of the existing code from UserForm_Activate, you must clear it and place only

Private Sub UserForm_Activate()
  DoEvents
  DotAnimation
End Sub

I was afraid of a really infinite loop and I limited it to 4 cycles. See the Static AnimNo variable. After testing, you can remove it, or extend the cycles number to whatever you need...

Theoretically, DoEvents should allow you to work with the form in parallel...