Excel VBA wildcard search

You can use Like operator (case-sensitive):

Private Sub search_Click()
    For firstloop = 3 To 10
        If Range("G" & firstloop).Text Like name.Text & "*" Then
            MsgBox "Found!"
            Exit Sub
        Else
            MsgBox "NOT FOUND"
        End If
    Next
End Sub

for case-insensitive search use:

If UCase(Range("G" & firstloop).Text) Like UCase(name.Text) & "*" Then

Also if you want to determine whether cell contains text (not only starts with text), you can use (case-sensitive):

If InStr(1, Range("G" & firstloop).Text, name.Text) > 0 Then

or (case-insensitive)

If InStr(1, Range("G" & firstloop).Text, name.Text, vbTextCompare) > 0 Then

UPD:

If the point only to show msgbox, then I'd suggest to use Application.Match:

Private Sub search_Click()
    If Not IsError(Application.Match("abc" & "*", Range("G3:G10"), 0)) Then
        MsgBox "Found!"
    Else
        MsgBox "NOT FOUND"
    End If
End Sub