Which is the largest Vaishnav Sampraday of Sanatan Dharma?
There are four Vaishnav Saṁpradāya namely;
Śrī Saṁpradāya (Ramavat or Ramanand Sampradaya) :- Param-Acharya : Bhagavati Śrī Sita Ji
Brahmā Sampradaya (Madhva Sampradaya) :- Param-Acharya : Brahmā Ji
Rudra Sampradaya (Vishnu-swami Sampradaya) :- Param-Acharya: Bhagavan Shankar Ji
Sanakadik Sampradaya (Nimbarka Sampradaya) :- Param-Acharya : Sanakadik Muni
And the division of Catuh (4) Sampraday is based of 52 Spritual Clans (52 dvare). Anyone who could not trace his descent back to the founder of a spiritual clan was not a member of any of the four sects of the catuh sampradaya. Of the fifty-two spiritual clans (52 dvaras), thirty-six were founded alone by Ramanandi ascetics and twelve were founded by Nimbarki ascetics. The remaining four spiritual clans were founded by ascetics of the Madhvacarya, and Visnuswami sects. The thirty-six founders of the Ramanandi spiritual clans fall within the first and sixth generations of Ramanand's descendants. Following Nabha Ji's list of twelve disciples, twenty-seven spiritual clans were founded by Anantanand and his descendants; four spiritual clans were founded by Sursurananda and his descendants. Śrī Sukhānand, Śrī Naraharyānand, Śrī Bhāvānandajī, and Śrī Pīpājī each founded a spiritual clan. Shri Ramanand Sampraday is the Largest & the Oldest Vaishnava sect in India with 36 Dvaras (spiritual clans) out of total 52 dvaras. In every meeting of the four bonafide Vaishnava-sects, Śrī Ramanand Sampradaya represents the Sri-Sect. Refer this official book of Catuh Saṁpradāya to know more.
Approx every Single Rishi-s, Kings of Sun & Lunar Dynasty were initiated in Śrī Saṁpradāya (which is also known as Śrī Rāmānanda Saṁpradāya, named after its 22nd Acharya) & got impression of Bow & Arrow and Shadakshar Śrī Rām Mantra in deeksha. As said in Sri Sada Shiv Samhitā;
नारदश्चाह वाल्मीकीं वाल्मीकिश्च लवं कुशम् । चन्द्रसूर्यादिवंश्याश्च नृपास्तेन सुदीक्षिताः ॥ श्रीराममन्त्रायुधाभ्यां च अङ्किताश्च शुभे भवे । हनूमांस्तु अगस्त्याय अगस्त्यश्च सुतीक्षणम् ॥ सुतीक्षणेन महाभागाश्चांकिता बहवः मुनीन् ।
Sri Narad Ji gave Śrī Rāma Mantra & bow and Arrow impressions to Sri Vālmīki than Śrī Valmiki gave it to Lava & Kusha. Than Sri Lava & Kusha gave Sri Ram Mantra & Bow and Arrow impressions to the Kings of Solar & Lunar Dynasty. And also Sri Hanuman Ji gave Sri Ram Mantra to Maharishi Agastya and Sri Agastya gave Sri Ram Mantra to Sri Sutīkshan Ji. Than Sri Sutīkshan Ji gave Shadakshar Śrī Ram Mantra & impressions of Bow and Arrow to many fortunate Rishi-s.
(Even Śrī Hanumān ji has given Mantra Dikshā to Vibhīshana also as said in Śrīmad Hanumān Chalisa, तुम्हरो मंत्र बिभीषण माना । लंकेश्वर भए सब जग जाना ॥)
(Quoted from Sri Sada Shiv Samhitā, Shloka 726-728 from Sri Ram Parattva by Śrī Brahma Dās in 1785)
Mentions of Catuh Sampradāya;
अतः कलौ भविष्यन्ति चत्वारः सम्प्रदायिनः ।
श्री माध्विरुद्रसनका वैष्णवाः क्षितिपावनाः ॥
रामानन्दो हविष्याशी निम्बार्कश्च महेश्वरि ॥
(Quoted from Padmā Purān mentioned in Shabdkalpadrum & Vachaspatyabhidhan)
रामानंद उदार सुधानिधि अवनि कल्पतरु।
विष्णुस्वामी बोहित्थ सिन्धु संसार पार करु॥
मध्वाचारज मेघ भक्ति सर ऊसर भरिया।
निम्बादित्य आदित्य कुहर अज्ञान जु हरिया॥
जनम करम भागवत धरम संप्रदाय थापी अघट।
चौबीस प्रथम हरि बपु धरे त्यों चतुर्व्यूह कलिजुग प्रगट॥
(Quoted from Sri Bhaktamal Chappay 28)