The library has no global state, so separate VmaAllocator objects can be used independently. There should be no need to create multiple such objects though - one per VkDevice is enough.
By default, all calls to functions that take VmaAllocator as first parameter are safe to call from multiple threads simultaneously because they are synchronized internally when needed.
When the allocator is created with VMA_ALLOCATOR_CREATE_EXTERNALLY_SYNCHRONIZED_BIT flag, calls to functions that take such VmaAllocator object must be synchronized externally.
Access to a VmaAllocation object must be externally synchronized. For example, you must not call vmaGetAllocationInfo() and vmaMapMemory() from different threads at the same time if you pass the same VmaAllocation object to these functions.