I owned VGTSX for years now but only in year 2011 was it a requirement for me to file taxes.
Can I claim foreign tax credit carryovers for past years even though I didn't file taxes during those years?
In order to be eligible for credits you must file a tax return. You should file the return even if no tax is due, and even if you otherwise are not required to file. That's the only way to secure credits. Since you didn't file prior year tax returns - there are no prior year credit carry-overs, because there are no prior year credits.