What are the obstacles to a quick survey of statin usage in COVID-19 patients?
The problem with your setup is that by targeting an ICU ward you are already narrowing your study to those who developed severe COVID-19. A more robust design would be a retrospective cohort study, where the exposure of interest has already been recorded.
E.g. the UK Biobank (~500,000 participants) has recently incorporated SARS-CoV-2 test results into its dataset, so you could use this to:

create a case/control split based on exposure (i.e. patients' historic use of statins)*
study the effect this has on outcome (i.e. development of severe COVID-19)


In  this  study,  we implemented rapid dynamic linkage, which allows us to provide a regular feed of new COVID-19 (SARS-CoV-2) test results to UK Biobank to facilitate rapid and urgent research into the epidemiological and human genetic risk factors for severe infection in the cohort.


Dynamic linkage of COVID-19 test results between Public Health England’s Second Generation Surveillance System and UK Biobank, Microbial Genomics (2020) **


* Or any of the other thousands of variables recorded in UKB.
** Disclaimer: I have worked on this project.