Пример запроса
Приложение загружает файл otpusk.avi в каталог /a/ на Диске пользователя, указывая контрольную сумму и хэш для проверки дубликатов.PUT /a/otpusk.avi HTTP/1.1
Host: webdav.yandex.ru
Accept: */*
Authorization: OAuth 0c4181a7c2cf4521964a72ff57a34a07
Etag: 1bc29b36f623ba82aaf6724fd3b16718
Sha256: T8A8H6B407D7809569CA9ABCB0082E4F8D5651E46D3CDB762D02D0BF37C9E592
Expect: 100-continue
Content-Type: application/binary
Content-Length: 103134024

Private Function Str2Byte(str$) As Byte()
    Str2Byte = StrConv(str, vbFromUnicode)
End Function
Private Function urlencode$(url)
    With CreateObject("scriptcontrol")
        .Language = "JavaScript"
        urlencode = .eval("encodeURI('" & url & "')")
    End With
End Function
Private Function MD5(ByVal bytes) As String
    Dim sTmp$, i%, byteArr() As Byte
    byteArr = bytes
    With CreateObject("System.Security.Cryptography.MD5CryptoServiceProvider")
        byteArr = .ComputeHash_2(byteArr)
    End With
    For i = 0 To UBound(byteArr)
        sTmp = sTmp & LCase(Right("0" & Hex(byteArr(i)), 2))
    Next
    MD5 = sTmp
End Function
Private Function Sha256(ByVal bytes) As String
    Dim sTmp As String
    Dim i As Integer
    Dim byteArr() As Byte
    byteArr = bytes
    With CreateObject("System.Security.Cryptography.SHA256Managed")
        byteArr = .ComputeHash_2(byteArr)
    End With
    For i = 0 To UBound(byteArr)
        sTmp = sTmp & LCase(Right("0" & Hex(byteArr(i)), 2))
    Next
    Sha256 = sTmp
End Function

Public Sub ÇàãðóçêàÔàéëà()
    Dim FileContents As Variant
    Dim FileName As String
    Dim stream As Object
    Dim http As Object
    Dim LocalFilePath As String
    Dim RemotePath As String
    
    LocalFilePath = "c:\#work\777(minus).mp3"
    RemotePath = "777.mp3"
    
    Host_disk = "https://webdav.yandex.ru/"
    RemotePath = Host_disk & RemotePath & "/"
    api_token = "e8cdc14be1604caeb4d2aac960a6027c"
    id_token = "e8cdc14be1604caeb4d2aac960a6027c"
        
    Set stream = CreateObject("ADODB.Stream")
    
    With stream
      .Type = 1
        .Open
        .LoadFromFile LocalFilePath
        FileContents = .Read
        .Close
    End With
    
    Set http_disk = CreateObject("WinHttp.WinHttpRequest.5.1")
    With http_disk
        .Open "PUT", urlencode(RemotePath)
        .setRequestHeader "Host", "webdav.yandex.ru"
        .setRequestHeader "Accept", "*/*"
        .setRequestHeader "Etag", MD5(FileContents)
        .setRequestHeader "Sha256", Sha256(FileContents)
        .setRequestHeader "Expect", "100-continue"
        .setRequestHeader "Content-Type", "application/binary"
         
         Êëþ÷ = "oauth_token=" & Chr(34) & api_token & Chr(34)
         Êëþ÷1 = " oauth_client_id=" & Chr(34) & id_token & Chr(34)
        
        
        .setRequestHeader "Authorization", "OAuth " & Êëþ÷, Êëþ÷1
        .setRequestHeader "Content-Length", UBound(FileContents) + 1
        .send FileContents
        .WaitForResponse
        Debug.Print .StatusText
    End With
End Sub