It is a matter of historical interest to find out the time frame of Sri Vaishnava System in Hindu Thought. And it would be great if one could find evidence, especially Epigraphs/Inscriptions.And it should lend itself to Dating. It also should meet up with the evolution of the script in which it is written.

Saivism and Vaishnavism, or for that matter any system of worship based on the God one worships, is found in the Vedas.Vedas speak of Reality,Brahman and also it has Hymns dedicated to various Deities like Agni,Indra,Varuna,Vishnu,Narayana,Sri,Durga ,Rudra,Maruts….But no specific instructions or rules are laid to declare one to belong to Either Saivism ,Vaishnavism etc

In fact Vedas speak of Nirguna Aradhana of Brahman,that is the worship or meditation of the Reality ,Brahman as One Beyond Attributes.However,at the practical level it was found difficult for a common man to concentrate on an Abstract Principle for Self Realisation,worship of various Deities evolved. This is called Saguna Aradhana or Worship of Brahman with Attributes.

As to when exactly it happened , one is not sure as Sanatan Dharma is not an Institutionalised Faith. It is a Way of Life and strictly Personal.

Even during the period of Mahabharata,there seems to have been no sects like Saivism, Vaishnavism.

The earliest inscription on this is found in Rajasthan.

Written in Brahmi script,it speaks of Krishna and Sankarshana and identifies them.

This has been dated to First Century BC.

The Epigraph also speaks of Pushyamitra as having performed Aswamedha Yaga.

So ,on this basis,one may say that the Vaishnava as a group might be dated to 1 Century BC.

But ,when one looks at the worship of Vishnu as Thirumal , Maayon, in Tamil Classics,and as the Date of Tamil city Poompuhar at 12,000 years ago, it might also be said that the separate groups called as Saivism, Vaishnavism might be dated to 10000 BC.

And if we take into account of Lemuria,which is dated around 230 Million years ago and MU civilization abutting Lemuria and Tamil was very much in evidence,we can date these groups to have originated around 200 million years ago.

The Hathibada Ghosundi Inscriptions, sometimes referred simply as the Ghosundi Inscription or the Hathibada Inscription, are among the oldest known Sanskrit inscriptions in the Brahmi script, and dated to the 1st-century BCE. The Hathibada inscription were found near Nagari village, about 8 miles (13 km) north of Chittorgarh, Rajasthan, India, while the Ghosundi inscription was found in the village of Ghosundi, about 3 miles (4.8 km) southwest of Chittorgarh. They are linked to Vaishnavism tradition of Hinduism.’

The discovered inscription is incomplete, and has been interpolated based on Sanskrit prosody rules.

Fragment A

Fragment A (Ghosundi stone inscription).

1 ….. tena Gajayanena P(a)rasarlputrena Sa-
2 ….. [j]i[na] bhagavabhyam Samkarshana-V[a]sudevabhya(m)
3 ……bhyam pujasila-prakaro Narayana-vat(i)ka.

Fragment B
1. ….[tr](e)(na) Sarvatatena As[v]amedha….
2 …..sarvesvarabh(yam).

Fragment C

Fragment C (Hathibada stone inscription)

1 ….vat(ena) [Ga]j(a)yan[e]na P(a)r(asaripu)t(re)na [Sa](r)[vata]tena As(vame)[dha](ya)- [j](ina)
2 ….(na)-V(a)sudevabh[y]a(m) anihata(ohyam) sa(r)v(e)[s]va[r](a)bh(yam) p(u)[j](a)- [s](i)l(a)-p[r]a[k]aro Nar[a]yana-vat(i)[k](a).

– Ghosundi Hathibada Inscriptions, 1st-century BCE

Extrapolation.

Bhandarkar proposed that the three fragments suggest what the complete reading of fragment A might have been. His proposal was:

Fragment A (extrapolated)
1 (Karito=yam rajna Bhagava)tena Gajayanena Parasariputrena Sa-
2 (rvatatena Asvamedha-ya)jina bhagava[d*]bhyaih Samkarshana-Vasudevabhyam
3 (anihatabhyarh sarvesvara)bhyam pujasila-prakaro Narayana-vatika.

– D. R. Bhandarkar’

Reference and citation.

1. Richard Salomon (1998). Indian Epigraphy: A Guide to the Study of Inscriptions in Sanskrit, Prakrit, and the Other Indo-Aryan Languages. Oxford University Press. pp. 239–240. ISBN 
2. ^ ^a b Theo Damsteegt (1978). Epigraphical Hybrid Sanskrit. Brill Academic. pp. 209–211.
3. ^ Jan Gonda (2016). Visnuism and Sivaism: A Comparison. Bloomsbury Academic. pp. 166 note 243. ISBN 978-1-4742-8082-2.
4. ^ James Hegarty (2013). Religion, Narrative and Public Imagination in South Asia: Past and Place in the Sanskrit Mahabharata. Routledge. pp. 46 note 118. ISBN 978-1-136-64589-1.
5. ^ ^{a b c d e f} D. R. Bhandarkar, Hathi-bada Brahmi Inscription at Nagari, Epigraphia Indica Vol. XXII, Archaeological Survey of India, pages 198-205
6. ^ Dilip K. Chakrabarti (1988). A History of Indian Archaeology from the Beginning to 1947. Munshiram Manoharlal. pp. 137–138. ISBN 978-81-215-0079-1.
7. ^ ^a b Gerard Colas (2008). Gavin Flood (ed.). The Blackwell Companion to Hinduism. John Wiley & Sons. pp. 230–232. ISBN 978-0-470-99868-7.
8. ^ Rajendra Chandra Hazra (1987). Studies in the Puranic Records on Hindu Rites and Customs. Motilal Banarsidass. pp. 200–201. ISBN 978-81-208-0422-7.
9. ^ Srinivasan, Doris (1979). “Early Vaiṣṇava Imagery: Caturvyūha and Variant Forms”. Archives of Asian Art. 32: 50–51. ISSN 0066-6637. JSTOR 20111096.