Are there any accredited universities in the world offering students the option of acquiring a BA/BSc in psychology completely online?
Are there any accredited universities in the world offering students the option of acquiring a BA/BSc in psychology completely online?


Yes

There are likely hundreds of programs. How well they meet your needs will be determined by 1) What you mean by 'accredited', and 2) residency restrictions.


  Accreditation


Are you interested in accredited institutions or accredited programs? Institutions are accredited by organizations recognized at the government level as being able to guarantee that all programs at the institution meet some minimum standards. In the US, regional accreditation is king, while the more prestigious sounding national accreditation is generally viewed as easier to get and less rigorous. In Canada, accreditation appears to be handled through membership in the Association of Universities and Colleges of Canada. 

Again, in the US, many of the large flagship state institutions have an online campus to accompany their bricks-and-mortar traditional campuses. For example Univerity of Maryland University College appears to offer an online BS in Psychology. UMUC is regionally accredited in the US by the Middle States Association of Colleges and Schools, just like the the bricks-and-mortar campuses of the University of Maryland system. Certainly, some of the larger schools in Canada also have a similar online presence. 

The on-line for-profit schools in the US tend to be nationally accredited, with the notable exception of the much maligned University of Phoenix, which offers an online program in psychology. University of Phoenix is regionally accredited by the Higher Learning Commission as part of the North Central Association of Colleges and Schools.

Individual programs may also be accredited or certified by various professional organizations. If you are looking for an online psychology program certified by a professional psychology association, try starting at the website of the association and look for lists of certified programs.


  Residency


Attending an online only institution should not have residency restrictions based on your country of citizenship, but it probably does. For example, I would imagine you would need to have a US education visa in order to be admitted into some US institutions, even if the program is entirely online and you never need to step foot on campus.