Newspapers reported last week that Facebook paid only £238,000 in corporation tax on UK earnings of up to £175m.
Coffee giant Starbucks has also come under fire for avoiding paying corporation tax on UK sales of £398m last year.
MPs are calling upon HM Revenue & Customs to launch an investigation.
Writing in his blog, CAP head Niall Cooper said he was considering organising a "Switch Off Facebook" day to force the social networking site to pay its fair share.
He put the proposal to his readers, with 93% saying they supported the idea.
He said: "Every pound Facebook and other major corporations dodge in taxes is a pound less to spend on essential public services, health, education and benefits – and the impact is increasingly being felt by individuals, families and communities across the country. Facebook generates the bulk of its revenue using a ‘pay per view’ or ‘pay per click’ advertising. If Facebook users publicly ‘switched off Facebook’ for a day – and at the same time called on the company to pay its fair share of taxes – we would hit them where it hurts most: In the wallet."