You can use the following code (FClient if object of class THttpClient):
uses
{…}, Sparkle.WinHttp.Engine, Sparkle.WinHttp.Api;
TWinHttpEngine(FClient.Engine).BeforeWinHttpSendRequest :=
procedure(Handle: HINTERNET)
begin
WinHttpCheck(WinHttpSetOption(Handle, WINHTTP_OPTION_PROXY_USERNAME, PWideChar(UserName), Length(UserName));
WinHttpCheck(WinHttpSetOption(Handle, WINHTTP_OPTION_PROXY_PASSWORD, PWideChar(Password), Length(Password));
end;