Short answer: under the FSF's definitions, I'd say the remote storage of encrypted passwords is too simple and too secured of an operation to count as problematic SaaS. However, modern password managers may do much more than this core functionality, which could run contrary to the FSF's recommendations.
Full analysis:
The FSF regards SaaS as an issue that is distinct from proprietary software:
SaaS and proprietary software lead to similar harmful results, but the causal mechanisms are different. With proprietary software, the cause is that you have and use a copy which is difficult or illegal to change. With SaaS, the cause is that you use a copy you don't have. [...]
Many free software supporters assume that the problem of SaaS will be solved by developing free software for servers. [...] But if the programs on the server are free, that doesn't protect you as the server's user from the effects of SaaS. They give freedom to the operator, but not to you.
The FSF's issues might be restated as
With SaaS, you give away your data to an external operator, and
You cannot modify a SaaS offering, even if the source code is free (instead, you must modify the free code and make your own service), so your computing done on someone else's service is done in a way that you cannot control.
However, the FSF makes it clear that not all network services are objectionable SaaS:
Does avoiding SaaS mean you refuse to use any network servers run by anyone other than you? Not at all. Most servers do not raise this issue, because the job you do with them isn't your own computing except in a trivial sense.
The original purpose of web servers wasn't to do computing for you, it was to publish information for you to access. Even today this is what most web sites do, and it doesn't pose the SaaS problem, because accessing someone's published information isn't a matter of doing your own computing. Neither is publishing your own materials via a blog site or a microblogging service such as Twitter or identi.ca. The same goes for communication not meant to be private, such as chat groups.
So, could the remote components of a password vault be said to be "doing your computing for you" in any substantial way? It seems like a password vault service has two parts:
A client to generate, encrypt, decrypt, and output your passwords
A service to store and serve encrypted passwords to the client
Functionality #1 has no network component, so we can be sure it is not problematic SaaS. #2 handles an extremely simply use case that neatly matches the FSF's acceptable use of a network service "publish[ing] information for you to access".
Functionality #2 needs only to store encrypted blobs of data. It is possible for a service to do this without giving away your sensitive data (since it is encrypted before submission) and without preforming any other computation beyond storing and re-serving that encrypted data upon request. In fact, these requirements are so easy to satisfy, they can be abstracted onto any arbitrary storage provider: one practical example is KeePass, a FOSS password manager, which supports a number of external services (Dropbox, OneDrive, as well as arbitrary network locations via FTP and SSH) to store and serve your encrypted passwords.
All of this means that remote storage and retrieval can be done in line with the FSF's recommendations: there does exist (both in theory and in actual fact) a password manager with remote storage that the FSF would find unobjectionable. However, it is also completely possible for a particular software/service to supply vault functionality in a way that does not conform to the FSF's recommendations. In particular:
Any storage model that does not let the user enforce privacy of their passwords with cryptographic certainty is worthless from a privacy perspective. If the service remotely holds or generates either cryptographic keys or unencrypted passwords, then there can be no guarantee that the service respects the privacy of your passwords.
A storage model that doesn't let you retain your passwords locally puts you at risk of the service failing to give you your own passwords one day (e.g., either because they went bankrupt or you didn't renew your subscription).