Yes, it's difficult indeed to provide any help without any more pieces of information...
You could make a full exchange log with the TIdLogFile class/component. Or look at the error returned by Indy (you can use the exceptions class for that).
Anyway, a few more things to check:
1/ Be sure to allow redirection, with something like (you may also fill the appropriate fields in you use a TIdComponent):
...
MyIdHTTP.HandleRedirects:=True;
MyIdHTTP.RedirectMaximum:=10; // (Generally, 3 is quite enough)
...
2/ If redirection is used, be sure that the HTTPS protocol is not required by the server. It's becoming more and more frequent that servers redirect automatically their clients to their encrypted version. If so, you'll have to add the 2 openssl .dll files.
3/ And finally, as an addition to Taazz's point 3, be sure to modify the by-default user agent of the Indy http component.
The by-default value is:
User-Agent: Mozilla/3.0 (compatible; Indy Library)
The problem is that this value is blacklisted by several domains/servers. AFAIK, the reason why is that Indy has been used in the past by malware programs (I'm not completely sure about the reason).
You can make a test with this Mozilla value for instance:
MyIdHTTP.Request.UserAgent:='Mozilla/5.0 (Windows NT 5.1; rv:29.0) Gecko/20100101 Firefox/29.0'; // Some web sites don't like the old by-default one (i.e. malware)