Well, yes, Windows Server 2008 is already very old, probably it doesn't support the modern TLS used by the server it's trying to access. Maybe the following information might help.
You must first make sure you have update 3140245 installed in Windows 7. See link below.
https://support.microsoft.com/en-ca/help/3140245/update-to-enable-tls-1-1-and-tls-1-2-as-default-secure-protocols-in-wi
Then, you can enable TLS 1.1 and 1.2 in WinHttp either from code, using Sparkle:
OptionValue :=
WINHTTP_FLAG_SECURE_PROTOCOL_TLS1_1 or WINHTTP_FLAG_SECURE_PROTOCOL_TLS1_2;
if not WinHttpSetOption(
TWinHttpEngine(Client.Engine).Session,
WINHTTP_OPTION_SECURE_PROTOCOLS, @OptionValue, SizeOf(OptionValue)) then
WinHttpCheck(False);
Or you can create specific registry keys to enable TLS1.1 and TLS 1.2 for all applications without you having to change its code. It’s described in the link above, you have to create the DefaultSecureProtocols registry value in either
HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Internet Settings\WinHttp
or
HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Internet Settings\WinHttp
depending on your OS version.