I - a private individual, unconnected with the civil aviation industry in any way except as an end-user - wish to obtain information about "a certain organisation" under the provisions of either the Data Protection Act or the Freedom of Information Act.
Is there anything in law that would prevent me paying that certain organisation the statutory fee and then demanding information on its financial situation?
As far as the Data Protection Act is concerned, nothing at all, but it wouldn't get you very far, since it only concerns personal data. You could pay your £10 for a subject access request, but the reply you would get would be along the lines of
"Dear Sir/Madam, thank you for your request. We have checked our records and can confirm that the personal data we hold about you is as follows:
No records found.
Yours faithfully"
As for the Freedom of Information Act, this applies to public authorities only and does not cover trades unions.