Is it ever possible to immigrate to Canada from an EU country professionally, if one is working in web design (websites, graphics, etc.)?
For Federal Skilled Workers Program (FSW), your work experience as a web designer is not in the 29 NOC list so you're not eligible for this one. http://www.cic.gc.ca/english/immigrate/skilled/complete-applications.asp
But you can try other programs to go to Canada as an immigrant.
1. Provincial Nominee Program
2. Quebec Skilled Worker Program (must know a little bit of French language)
3. Arranged Employment Opinion (AEO)
4. PhD study Program (this is new)