Does MSR apply to online sales? Yes. Products offered online or via distance sales are deemed to be made available on the EU market if the offer is targeted at end users in the Union (Article 6).
Do we need an EU 'responsible person' or economic operator? For products covered by the laws listed in Article 4(5), the product may be placed on the market only if there is an EU-established economic operator responsible for Article 4 tasks (Article 4(1)-(2)).
What will authorities ask for first? Product identification + operator contact details, declarations (DoC/DoP), and an index to technical documentation and tests - plus what corrective actions you are taking.