How to save illustrator file as jpeg for print design

as  you  suggested … you  may  purposely   'contact'  the  website  proprietor … encourage  them  to  start  supporting  *.pdf  format  or  even  *.svg  format … however,  pdf  will  be  a  more  likely  candidate … as  svg  favors  interactive/ animation  elements.

looks like you've just drawn it really small.  you can just go to export for web, and then choose a size such as 2500px high and you'll get a better quality image out, but your artboard doesn't match the design well so you'll get a load of white space to the right of the donut. You could use the artboard tool as you have currently selected in that screen shot, to make a better shape first (maybe squarer).

Ideally you'd adjust the artboard shape and make it larger then scale your object up to fill it before exporting it. Or you can just select your design, 'Edit > copy', then open a new A4 sized doc and paste it in, then scale it up to fill the page then when you then export as jpeg it will be a much high resolution image.

You can try:

File > Export > Export As > JPEG

Check Use Artboards (if your Artboard matches the size of your card).

Choose High (300ppi) for Resolution and Art Optimzed for Anti-aliasing.

JPEG is a raster image. As well as PNG. So you can expect pixelization with those two.

Your only chance would be to make it bigger than you need and then scale it down.

If PDF will be accepted by the service as well, try that.