Can we chant Vishnu Sahasranamam daily without initiation?
I don't think Initiation is needed for chanting Vishnu Sashranama. I will keep my personal opinion aside here and go strictly by scriptures.
Of course initiation in needed to chant vedic mantras. Since vedas cannot be understood by less intelligent people like us, Puranas and Itihas were written down so that essence of vedas can be taught to people through stories.
Here is a statement from Srimad Bhagvatam-
Thus the great sage Vyāsadeva, who is very kind to the ignorant masses, edited the Vedas so they might be assimilated by less intellectual men. Out of compassion, the great sage thought it wise that this would enable men to achieve the ultimate goal of life. Thus he compiled the great historical narration called the Mahābhārata for women, laborers and friends of the twice-born.
Source : Srimad Bhagvatam Canto 1 chapter 4
As you can see Mahabharata was clearly written for less intelligent people and even for people who cannot go through the process of initiation.
Vishnu-Sahasranama itself is a part of Mahabharata, hence everyone is allowed to chant it.