No, Nama Japa does not need initiation.
We accept Shastravakya and Aptavakya as pramana. Also, "Yukiheenavicharaena dharmahaanih prajayate" which means logical analysis is accepted in our spiritual culture. (Reference: Yogavaasistha-Ramayana, quoted in Jnaniguru, Nigamananda Saraswati, page 12).
Remembering the above, let us proceed:
Naama Japa and Sankirtana was popularised by Mahaprabhy Sri Chaitanyadeva. He says in Sri Chaitanya-Charitamrita (Chapter 15):
এক কৃষ্ণনাম করে সর্বপাপক্ষয়।
নববিধ ভক্তিপূর্ণ নাম হইতে হয়॥
দীক্ষা পুরশ্চর্যা বিধি অপেক্ষা না করে।
জিহ্বাস্পর্শে আচণ্ডাল সবারে উদ্ধারে॥
"Nama of Krishna alone is enough to destroy all sins and attainment of matured levels of the nine types of devotions. It does not need initiation(Diksha) or Purascharana. Just chanting of it liberates all men of all castes.
This we can of course take as Aptavakya."
Coming to the scriptures, The Vrihat-Naradiya-Purana (38/126) announces:
हरेर्नाम हरेर्नाम हर्र्नामैव केवलम्।
कलौ नास्त्येव नास्त्येव नास्त्येव गतिरन्यथा॥
The nama of Hari, the nama of Hari, the nama of Hari alone [is the way]. In the age of Kali there is no other method of Liberalisation.(thrice).
The words "Kevalam" and "No other way" means no anusthanas like Diksha is necessary.
Now the logical part. Did Mahapraprabhu exclude the names of other forms of God like Shiva, Shakti etc? No. Because in His Shikshastakam, He writes:
नाम्नामकारि बहुधा निजसर्वशक्तिस्तत्रार्पिता नियमित: स्मरणे न कालः।
"[O God,] You have assumed different names and have poured all your power in each and there is no particular time of remembering it.
So quite definitely He meant names of all forms of God. This view was accepted by Mahajanas like Sri Ramakrishna (Ref: Sri Ramakrishna Kathamrita), Sri Charandas Babaji, Sri Ramdas Babaji(Ref: Their biographies by Abha Sarkar), Sri Sitaramdas Omkarnath(Omkarnath Rachanavali) and many other realised souls.
There are scriptural proofs here also. In Shivarahasya, we get
तारकं ब्रह्मपरमं शिव इत्यक्षरद्वयम्
The nama of Shiva is the TAraka Brahma. (Reference: Shivanamamritalahari, Sri Sitaramdas Omkarnath, OmkarnathnRachanavali, Vol.11, page 15).
In Mundamalatantra, we get
यः जपेत्सततं दुर्गां यो जपेत् परमं मनुम्। स जीवलोके देवेशि नीलकण्ठत्वमाप्नुयात्॥
Whoever remembers Dura or makes japa of Durganama constantly, becomes Shiva Himself. (Reference: Omkarnath Rachanavali, Chapter 11, page 192).
Hence it is proved that nama-japa does not need initiation.