Certificates are transient in nature: they expire, and must be renewed. Even worse, the validity of a certificate is the property of the current time, since certificates may be revoked at any time. Therefore, if you want to store signed documents, and be able to validate them at a later date, then you need time stamps. See this answer for some details.
While time stamps protect your signed documents against a cessation of activity from the CA, an abrupt closure would still be a problem since, by ceasing to issue CRL on a regular basis, the CA effectively revokes all existing certificates (at least that's the effect from the outside). Normally, contracts with CA include strict clauses on the continuity of operations; up to an including financial provisions and/or to guarantee that even in case of total business failure, CRL will still be issued for some years.
When you renew your certificate, you are actually acquiring a new one. Nothing forces you to get that second certificate from the same source as the other one. You may think of these certificates as "the old one" and "the new one", but from an X.509 point of view they are just two certificates, that may relate to distinct issuing CA. You will have to renew certificates on a regular basis anyway, since they have expiration dates. An out-of-business CA may "just" force you to renew earlier than expected.
Alternatively, you may want to operate your own CA, but it is going to be very expensive if you want to do it properly (a serious CA is 5% technology, 95% people and procedure).