>>> Individuals can't ask for a DBS check unless they are under the umbrella of an organisation or employer.
That ceased to be true when 'basic' DBS checks were introduced in England and Wales at the start of last year. (They'd been available in Scotland for many years before then). Anyone is now able to apply online for the results of such a check on themself. Similarly, any employer can now require a DBS check on applicants, irrespective of the type of work the job involves. (Previously only checks could only be carried out on applicants applying for roles which were exempted from the provisions of the Rehabilitation of Offenders Act). However 'basic' checks only show convictions which aren't spent. (So it doesn't prove that you've not fallen foul of the law at an earlier time).
If you need a document which shows all your convictions, or the total absence of any such convictions, you can make a subject access request to your local police force. (The police are entitled to charge you a tenner for the service). That will show all of the information the police hold about you (possibly other than intelligence relating to current investigations), including all of your criminal convictions (or their absence). Employers are not permitted to demand that job applicants present them with the results of a police subject access request, as to do so would effectively be getting an applicant to reveal any spent convictions.
You can also obtain a copy of the information held about you on the Police National Computer, free of charge:
https://www.acro.police.uk/subject_access.aspx
(Again, employers can't demand that you present them with a copy of such information)
If you need a copy of your police record for visa purposes then a police subject access request or a copy of your PNC data won't be sufficient. You'll need a Police Certificate, which costs £45:
https://www.acro.police.uk/police_certificates.aspx#