How to take ownership of an AWS tenancy that is managed by a third party who has gone out of business

Hello.

To update billing information, etc., you will need a user called the root user who can perform all operations on the AWS account.
To sign in as the root user, you must use the email address associated with your AWS account.
There is no problem if this email address is managed by your company, but if it is managed by a bankrupt third-party company, there is a high possibility that you will not be able to sign in.
https://docs.aws.amazon.com/signin/latest/userguide/introduction-to-root-user-sign-in-tutorial.html

By the way, depending on the content you want to change, even IAM users can perform the operation.
However, this requires IAM users to enable the necessary settings to access billing information.
https://docs.aws.amazon.com/awsaccountbilling/latest/aboutv2/manage-payment-method.html
https://docs.aws.amazon.com/awsaccountbilling/latest/aboutv2/control-access-billing.html

If you are able to sign in to your AWS account as an IAM user, I recommend that you contact AWS Support.
Inquiries under "Account and billing" can be made free of charge.
https://docs.aws.amazon.com/awssupport/latest/user/case-management.html

Just to add to the excellent answer from @Riku_Kobayashi is it the third-party that is the billing contact for the account, so they get the invoices from AWS every month and it's them that pays AWS? If it is, and they cease trading, it's very likely the account will be suspended due to the unpaid bills, and so the workloads that run your business will stop running.

Does the third-party have Enterprise support (either On-Ramp or full fat Enterprise)? If they do then they will have access to a Technical Account Manager who would be your first point of contact if this scenario unfolded. It would be advantageous to find out who this is in advance.