to answer directly the topic question: Yes, they do.
But to answer directly a concern: It is not on the security/encryption side.
There are a lot of forum posts where it is at first unclear wether the people talk about the mobile applications or the ones for the desktop. But to actually provide help this is a crucial fact. Also there exist many questions why does this work on the desktop and not my smartphone. This thread wants to summarize the differences so that you directly know how Cryptomator works on your platform.
What is their difference?
Apart from the obvious OS question, it is how they make your encrypted files accessible.
When you unlock a vault in the Cryptomator desktop application and browse through its content, you do it with the system file browser. This means that every request to read, write or move a file/directory is generated by the OS, then channeled through a “Cryptomator layer” and finally executed by the OS again.
Therefore the general rule of thumb is that the vault must be at least accessible by your operating system. There are exceptions, thou. But if your vault is stored on your hard drive, it definitely works. The OS capabilities to access remote files can be extended by third parties, e.g. MountainDuck, but it might introduce incompabilities.
On the mobile side everything works over the app, there is no system integration.
But more importantly your vault is not stored on your smartphone or tablet. Cryptomator connects with your cloud provider, queries the encrypted directories and shows a locally decrypted view. Also the content of a file is only downloaded on demand, i.e. when you want to view or edit it.