The advertising ecosystem has been progressively reshaped by computational advertising models facilitated and controlled by digital media platforms. Meta and Alphabet, two of the dominant players in this ecosystem, have contributed significantly to industry disintermediation via the design and market dominance of their advertising systems, which in turn are symbiotic with their own business models, and feed on large amounts of consumer data generated by everyday platform use. The operations and impacts of these systems remain opaque and difficult to observe. Responding to this challenge, we present a methodological framework integrating cultural, legal, media, and computational perspectives to enable the observation of computational advertising. This methodology comprises three novel computational methods: direct data gathering from ad transparency libraries; automated data donation via participatory citizen science browser plug-ins; and a privacy-aware mobile screen capture app. We describe how our suite of computational approaches enables scalable and participant-situated research, overcoming limitations posed by restricted platform access to computational advertising data. Ethical considerations and practical implications are discussed alongside case study findings, further demonstrating the potential for these methods to inform regulatory debates and scholarly advancements in the field.