How to get a vCard (.vcf file) into Android contacts from website

AFAIK Android doesn't support vCard files out of the Box at least not until 2.2.

You could use the app vCardIO to read vcf files from your SD card and save to you contacts. So you have to save them on your SD card in the first place and import them afterwards.

vCardIO is also available trough the market.

Just to let you know: I just tried it using a vCard 2.1 file created according to the vCard 2.1 spec. I found that vCard 2.1, despite being an old version, already covered everything I needed, including a base64-encoded photo and international character sets.

It worked perfectly on my unmodified Android 4.1.1 device (Galaxy S3). It also worked on an old iPhone 3GS (iOS 5, via the Evernote app) and a coworker's unmodified old Android 2.1 device. You only need to set the Content-disposition to attachment as suggested above.

A minor problem was that I triggered the VCF download using a QR code, which I scanned with the Microsoft Tag app. That app told me Android couldn't handle the text/x-vcard media type (or just text/vcard, no matter). Once I opened the link in a Web browser (I tried Chrome and the Android default browser), it worked fine.