Mantras of Krishna and Vishnu which does not require any initiation, does not have any rules to follow, can be chanted anytime and anywhere. Please don't include Hare Krishna Maha mantra in your answer as it requires initiation according to Radha Tantra. All the mantras should be backed by scriptural references, download links (if any). A humble request to people who are Krishna and Vishnu devotees or people knowledgable of mantras to answer this question. Many thanks in advance.
There are no such mantras. All mantras require initiation from a recognized Parampara, as chapter 10 of the Garga Samhita says:
Vamana, Brahma, Ananta Sesha and Sanaka Kumara will appear as brahmanas by the order of Visnu, for the preservation of eternal righteousness in kali yuga. Visnu Swami, Madhvacarya, Ramanuja and Nimbaditya will appear respectively as a portion of Vamana, Brahma, Ananta Sesha and Sanaka Kumara. These saviours will be the establishers of the four authorised and empowered spiritual channels of disciplic succession in the period calculated from the reign of King Vikrama in 54 B.C. subsequently through the present era of kali yuga. These four authorised and empowered spiritual channels of disciplic succession are to be fully accepted by all beings; as any word, combination of words or formulation of sounds frequencies, invoked or addressed, audible or inaudible, secret or revealed, ancient or contemporary, outside their auspices prove to have absolutely no efficacy.
There are thousands of Mantras in the Vedas, but the kind of Mantras you are looking for are not to be found in them.
Agamas or Tantras have plenty of them.
Bu for the Tantric Mantras Guru Diksha is required.
Lord Shiva says:
TasmAt siddhantam samprApya sampradAyAdihetubhih |
AntarenopodeshtAram mantrAh syur nishphalA yatah ||
KulArnava Tantram 14.4
It basically means, "without Guru Upadesha, all traditional knowledge, philosophies and Mantras fail to bear fruit".
I find many Mantras (like we discussed in chat) of Vishnu and Krishna in the Tantrasara. They are from SanatkumAra Tantra, Gautamiya Tantram, NArada PancharAtram etc.. So, they will all require initiation.
Also, as shown in this answer, chanting a Tantric Mantra without proper initiation is condemned:
Kalpe drishtA tu yo mantram japed gurumanAsritaha |
SutanAsho bhavet tasya phalam kinchinna vidyate ||
The person who, without depending on a Guru ( Gurumanasitaha), chants mantras by taking them from a book, gets his progeny destroyed [by doing so] (sutanasho) and does not get any benefits [of the chant] either.
Now, the Puranas also have few Mantras. What about them?
From the Devi Bhagavata PurAna's last chapter of the last Book:
So for the satisfaction of one’s own S’akti, all can read this, without contradicting each other. Never any woman nor any S’ûdra, is to read this herself or himself, even out of ignorance; rather they should hear this from the mouth of a Brâhmana. This is the rule of the S’âstras.
So, what it says those who do not have Vedic initiation, should not read the Purana and thus they can not chant the Puranic Mantras as well.
Similarly, the VyAsa Smriti says:
Brahmanas, Kshatriyas, and Vaishyas are called the. twice-born. Their first births take place when they a^re delivered of their mother's womb ; their second,when they duly accept the Gayatri Mantra from their preceptors. (21)
Thus made twice-born, and free from all other faults, they become entitled to study the Vedas, Smritis and Puranas. (22)
So, what it says, even for reciting the Mantras mentioned in the Puranas, one needs to get a Vedic initiation. And for Agamic Mantras a Guru Upadesha will be required.
So, i am afraid, no such Mantras are there.
The Hare Krishna Mahamantra prescribed by Mahaprabhu does not need initiation or purascharan etc but gives whatever one desires, according to Chaitanyabhagavat and Chaitanyacharitamrita. Mahaprabhu Himself declared this and this is supported by the scriptures like the kalisantaranopanishad etc