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The Vedas (Samhitas) do not talk about Sannyasa at all, so they won't have them.
The principle Upanishads (13/14 of them) also do not have them, but they do mention the word "Sannyasa".
Then, in which scriptures are the rituals and Mantras, to be used during initiation into Sannyasa, found?
Also, what Mantras are used and what is the process in brief?
Self answering... also a partial answer.
The process of taking Sannyasa is found in the book called the "Essence of Dharma Sindhu" hosted at the Kamakoti.org. But there is no mention of any scriptures from which the guidelines, Mantras etc are taken. Otherwise, the details is quite extensive.
Sanyaasa Grahana Vidhi:
Uttaraayana is stated to be the preferred time for initiating into Sanyasa, especially for those who have Grihyaagni. Having located a Sadguru who possesses Shanti and Danti o r Placidity and Self Restraint, the aspirant Sanya si should learn Sanyasi Dharmas from the tutelage and keep on practising Gayatri Japa, Rudra Japa and Kushmanda Homa for three months and after attai ning Bahyaantara Shuchi on one Rikta Tithi make th e Sankalpa: Amukasya Mama karishya -maana Sanyaase neydhikaaraartham Chatuh krucchaatmakam praayaschi ttam Pratikruccham tatprachyamnaayaika Gonishkriya dwaaraaha maacharishye Krucchaprtayaamnaaa Gonishk raya dravyam Viprebhyo daatumahamtsrujey/
Thus hav ing given Go-Pratyaamnaaya,commence series of Shra addhas on Ekadashi or Dwadashi at Brahma Ratri. As hramasastha has to perform ChatuhKruccha followed by Ashta Shraddhas; Aapastambas and Hiranyakeshiya s follow the Saankalpa Vidhi without Agnoukarana P indaas and by Sankalpa Vidhi only. But Ashwalaayan aadis follow Sapinda Paravana Prayoga; in this in Savya position they perform Shraddhaanga Tarpana with Yava mishra jala in rivers and waterflows sta ting: Brahmaanam Tarpayaami, Vishnum--; Maheshwara m--; Devarsheen--; Brahmarsheen--;Kshatrarsheen--; Vasoon--;Rudraan--;Adityaan--; Sanakam--;Sanandan am--; Sanaatanam--;Pancha Maha Bhutaani--; Chakshu raadi Karanaani--;Bhutagraamam--; Pitaram--; Pitaa maham--;Prapitaamaham--; Aatmaanam-/
Having return ed home the Karta would then announce Desha Kaalaa di and make the Sankalpa: Karishyamaana Sanyaaasaa ngatvey noushtaa shraadhhani paarvana vidhinaannen aameynavaa karishye/
This would be like Naandi Shr aaddha and hence in Savya position itself by using yavaas in place of Tilas. The opening Shraaddha is meant for Satya Vasu Sanjnika Vishwa Devas involv ing two Naandi Mukha Brahmanaas and naming (Vaana) of one of them to organise the Eight Shraaddhas. In the Prathama Deva Shraaddha, the ‘Ucchaarana’ or recital would be Brahma Vishnu Maheshwaraa Naan di Mukhaah Sthaanekshanah priyataam/ The Second one is Rishi Shraaddha with the Uccharana of Deve shi Brahmarshi Kshatrarshi yo naandi –mukhaah Stha anekshana priyataam/ The third one is Divya Shradd ha addressed to Vasu Rudraa- ditya Rupaa naandhi mukhaaya--/ The fourth one is Manushya Shraddha to Sanaka Sanandana Sanaatanaadi----; the fifth one is Bhuta Shraaddha to Prithivyaadi pancha Mahaa bh utaan –yekaadasha chakshuraadi karanaadi chaturvid ha bhutagraamaa naandi mukha;the sixth one is Pitr a Shraaddha to Pitru Pitaamaha Prapitaamahi naandi mukha----; the Seventh Shraaddha is directed to M aatru Pitaamahi Prapitaamahyo naandi mukha---; and finally the Eighth Shraaddha being Atma Shraaddha and the Uccharana is: Atmaantaraatma Paramaatmaa nah Naandi mukha----. After the Naandimukha, Padya is given to Vishwa De vas with the Mantra : Sanyaasaartha maham Shraaddh am kurvey bruta Dwijottamaah, Anugnaam praapya ush maakam siddhim praapyaami shasvateem/
Then one Vishwa Deva Patra and Ashta Patras for are arranged for Deva-Rishi-Divya-Manushya-Bhuta-Pitru-MaatruAtma to provide Aasana-Gandhaadi Ahutis are offere d. This method is by way of Sankalpa in the case of Apastambaas but Ashvaayanas perform with ‘Arghy apaatraa- sadana’. Pinda-daanaadikas also are not applicable to Apastambaas. After the Ashta Shraddh a on the first day, the Sanyasa Karmas on the foll owing day include kesha khandana , Nadi Snaana, ko upeena dharana, danda dharana, kamandalu dharana, Achhadana Vastra dhaaranaadi karmas. Then the Sanyasa Grahana Karya is followed by the Maha Sankalpa : Asesha duhkha nivrutti niratishaya -ananda pra apti rupa Parama Purushaartha praaptaye Parama Ham sa Sanyaasa grahanam karishye/ Tadangatayaa Ganapa ti pujana punyaahvachana Maatrukaa pujana Naandhi Shraaddhaani karishye/
The procedure is as follows : Recite the Mantra viz. Brahmaney namah, Vishna vey namah, Rudraaya namah, Suryaaya, Somaaya, Atm aney, Antaratmaney, Agnimeeley, Ishetvorjetwaa, Ag na Aayaahi Shannodevi etc.; take three fistfuls of ‘Satthu Pishta’ or mix of cereals, millets and pu lses, eat the Sattu thrice and touch the naabhi ; recite again the Mantra: Atmaney swaha Antaraatma ney swaaha Paramatmaney Prajaapataye swaaha; eat the mix of milk-curd-ghee-water by the Mantras of t rivridasi once, pravridasi twice, divrudasi thrice , then drink water saying Aapah punantu and finally state: Upavaasam karishye/
Thereafter, in that order, are to be performed the following rituals:
Savitri Pravesha
The Karta would recite : Om Bhuh Savitrim pravishaami; Om Tat Savaturvarenyam Om B huvah Savitrim pravishaami; Bhargodevasya dheemah i; Om Swaha Savitrim pravishaami; Dhiyoyonah prach odayat Om Bhurbhuvaswah Savitrim praveshayaami; Ta tsavatur varenyam Bhargo Devasya dheemahi dhiyoyon ah prachodayat/
Viraja Homa
This is a highlight Homa in the conte xt of Sanyasa Grahana Vidhana and is more or less the conclusive component of the Vidhana. Vira ja Mantra : It is with this Mantra that forty clim actic Ahutis to Agnideva are executed with a wide variety of Dravyas
Sarva Tyaga Vidhi:
Total Renunciation starts with the recital of the Maasa-Tithi-kaalas by the Karta as he would leave three handfuls of water in a Jalaashaya : Aparoksha Brahmaavaaptaye Sanyaasam Karomi/--Yeshahavaa Agnissurayah praanam gacchasw aa, Om swaamyonimgaccha swaahaa/
Viraja Homa
and finally, the Praisocchara:
Praishocchaara: The terminal acts of Sanyasa Vidha ana of Praishocchaara of the new Sanyasi are as f ollows: First with leaving water in the Flow with the Mantras thrice in three tones of low-medium an d loud voice: Om Bhuh Samnyastam mayaa, Om Bhuvah Samnyastam mayaa, Om Swaha Samnyastam mayaa, Om B hurbhuvaswah Samnyastam mayaa/ Then the Shikha is unfurled and cut; the Yagnopaveeta is taken out b y the raise of his hands above the shoulders and k ept in his hands reciting the Mantra:
For all these above processes, i have omitted a huge amount of actual details, because otherwise, it makes the answer exceptionally long.
Those who want to read in details can read this chapter of the book "Essence of Dharma Sindhu"
But, i am interested to know from which scriptures all these Mantras and procedures are taken. So, that part of the question is yet to be answered.
This is a partial answer.
Mantras for dasnami sannyasis are the four maha vakyas of the Upanishads. The mantra of the Puri Dasnami Sannyasis to which Ramakrishna Order belongs is
Ayamatma Brahma
Mandukya Upanishad 2
This is what I was told by Swami Sarvadevananda.