ICANN would appoint a receiver for the registrar's domains and the DNS would be continued until it is settled. This has happened before with a much smaller registrar.
I would expect 60 - 90 days pandemonium but the DNS disruption would be nil -- the domains' DNS entries would be frozen in the root servers and would not update.
Renewals and suspensions would be frozen until there was a back end interface made available to domain holders -- a moratorium period would be set on suspensions for non renewal until the interface is made available (I hope
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Otherwise, riots in the streets and an assault on the ICANN offices by litigation attorneys.