I have a Vista computer and cannot download Connectify as it requires Windows 7. Is it possible for me to connect to the Internet via Wi-Fi and share that connection through the same Wi-Fi adapter in my laptop?
This depends on the exact hardware, but the odds are strongly in favor of 'no'. Almost all wireless adapters only support a single concurrent connection; this would require two (one for the normal network, one for the ad-hoc network).
You could add a second wireless adapter, though.